Theorem fldextrspunlsplem 34103

Description: Lemma for fldextrspunlsp 34104: First direction. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypotheses

Ref	Expression
fldextrspunfld.k	⊢ 𝐾 = (𝐿 ↾s 𝐹)
fldextrspunfld.i	⊢ 𝐼 = (𝐿 ↾s 𝐺)
fldextrspunfld.j	⊢ 𝐽 = (𝐿 ↾s 𝐻)
fldextrspunfld.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspunfld.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspunfld.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspunfld.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspunfld.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldextrspunlsp.n	⊢ 𝑁 = (RingSpan‘𝐿)
fldextrspunlsp.c	⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
fldextrspunlsp.e	⊢ 𝐸 = (𝐿 ↾s 𝐶)
fldextrspunlsp.1	⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
fldextrspunlsp.2	⊢ (𝜑 → 𝐵 ∈ Fin)
fldextrspunlsplem.2	⊢ (𝜑 → 𝑃:𝐻⟶𝐺)
fldextrspunlsplem.3	⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿))
fldextrspunlsplem.4	⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))

Assertion

Ref	Expression
fldextrspunlsplem	⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))

Distinct variable groups:   𝐵,𝑎,𝑏,𝑓   𝐹,𝑎,𝑏,𝑓   𝐺,𝑎,𝑓   𝐻,𝑎,𝑏,𝑓   𝐽,𝑏   𝐾,𝑎,𝑏,𝑓   𝐿,𝑎,𝑏,𝑓   𝑃,𝑎,𝑏,𝑓   𝑋,𝑎   𝜑,𝑎,𝑏,𝑓

Allowed substitution hints:   𝐶(𝑓, 𝑎, 𝑏)   𝐸(𝑓, 𝑎, 𝑏)   𝐺(𝑏)   𝐼(𝑓, 𝑎, 𝑏)   𝐽(𝑓, 𝑎)   𝑁(𝑓, 𝑎, 𝑏)   𝑋(𝑓, 𝑏)

Proof of Theorem fldextrspunlsplem

Dummy variables 𝑐 𝑢 𝑒 ℎ 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
2	1	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐺 ∈ (SubDRing‘𝐿))
3		fldextrspunlsp.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
4	3	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
5		eqid 2765	. . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿)
6		fldextrspunfld.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Field)
7	6	flddrngd 20871	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
8	7	drngringd 20865	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
9	8	ringcmnd 20392	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ CMnd)
10	9	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ CMnd)
11		fldextrspunfld.6	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
12	11	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐻 ∈ (SubDRing‘𝐿))
13		sdrgsubrg 20924	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
14	1, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
15		subrgsubg 20706	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
16		subgsubm 19239	. . . . . . . 8 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → 𝐺 ∈ (SubMnd‘𝐿))
17	14, 15, 16	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubMnd‘𝐿))
18	17	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐺 ∈ (SubMnd‘𝐿))
19		eqid 2765	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
20	14	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐺 ∈ (SubRing‘𝐿))
21		fldextrspunlsplem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:𝐻⟶𝐺)
22	21	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑃:𝐻⟶𝐺)
23		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ 𝐻)
24	22, 23	ffvelcdmd 7084	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑃‘𝑓) ∈ 𝐺)
25		fldextrspunfld.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
26		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐼) = (Base‘𝐼)
27	26	sdrgss 20926	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ⊆ (Base‘𝐼))
28	25, 27	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐼))
29		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐿) = (Base‘𝐿)
30	29	sdrgss 20926	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
31	1, 30	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
32		fldextrspunfld.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (𝐿 ↾s 𝐺)
33	32, 29	ressbas2 17316	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
34	31, 33	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
35	28, 34	sseqtrrd 3975	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐺)
36	35	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
37	3	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
38	25	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐼))
39	11	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐻 ∈ (SubDRing‘𝐿))
40		ovexd 7451	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝐹 ↑m 𝐵) ∈ V)
41		simpllr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
42	39, 40, 41	elmaprd 33072	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
43	42, 23	ffvelcdmd 7084	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓) ∈ (𝐹 ↑m 𝐵))
44	37, 38, 43	elmaprd 33072	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓):𝐵⟶𝐹)
45		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑐 ∈ 𝐵)
46	44, 45	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐹)
47	36, 46	sseldd 3939	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐺)
48	19, 20, 24, 47	subrgmcld 33591	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) ∈ 𝐺)
49	48	fmpttd 7114	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
50	49	adantlr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
51		fveq2 6885	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑃‘𝑓) = (𝑃‘ℎ))
52		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑢‘𝑓) = (𝑢‘ℎ))
53	52	fveq1d 6887	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘ℎ)‘𝑐))
54	51, 53	oveq12d 7434	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
55	54	cbvmptv 5217	. . . . . . 7 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
56		fvexd 6900	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (0g‘𝐿) ∈ V)
57		ssidd 3961	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐻 ⊆ 𝐻)
58		fldextrspunfld.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
59		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
60	59	sdrgss 20926	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
61	58, 60	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
62	29	sdrgss 20926	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
63	11, 62	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
64		fldextrspunfld.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (𝐿 ↾s 𝐻)
65	64, 29	ressbas2 17316	. . . . . . . . . . . . 13 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
66	63, 65	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
67	61, 66	sseqtrrd 3975	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
68	67, 63	sstrd 3948	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
69	68	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐹 ⊆ (Base‘𝐿))
70	3	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
71	58	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐽))
72		ovexd 7451	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐹 ↑m 𝐵) ∈ V)
73		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
74	12, 72, 73	elmaprd 33072	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
75	74	ffvelcdmda 7083	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
76	70, 71, 75	elmaprd 33072	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ):𝐵⟶𝐹)
77		simplr 781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝑐 ∈ 𝐵)
78	76, 77	ffvelcdmd 7084	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
79	69, 78	sseldd 3939	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
80	21	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑃:𝐻⟶𝐺)
81		fldextrspunlsplem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿))
82	81	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑃 finSupp (0g‘𝐿))
83	8	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝐿 ∈ Ring)
84		simpr 490	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝑦 ∈ (Base‘𝐿))
85	29, 19, 5, 83, 84	ringlzd 20404	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → ((0g‘𝐿)(.r‘𝐿)𝑦) = (0g‘𝐿))
86	56, 56, 12, 57, 79, 80, 82, 85	fisuppov1 33075	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))) finSupp (0g‘𝐿))
87	55, 86	eqbrtrid 5148	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) finSupp (0g‘𝐿))
88	5, 10, 12, 18, 50, 87	gsumsubmcl 20013	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) ∈ 𝐺)
89	88	fmpttd 7114	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))):𝐵⟶𝐺)
90	2, 4, 89	elmapdd 8840	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) ∈ (𝐺 ↑m 𝐵))
91		breq1 5114	. . . . . 6 ⊢ (𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) → (𝑎 finSupp (0g‘𝐿) ↔ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿)))
92	91	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑎 finSupp (0g‘𝐿) ↔ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿)))
93		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))))
94	93	fveq1d 6887	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → (𝑎‘𝑏) = ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏))
95		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))
96		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑏 → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘𝑓)‘𝑏))
97	96	oveq2d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))
98	97	mpteq2dv 5207	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))
99	98	oveq2d 7432	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
100		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
101		ovexd 7451	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))) ∈ V)
102	95, 99, 100, 101	fvmptd3 7017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
103	102	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
104	94, 103	eqtrd 2800	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → (𝑎‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
105	104	oveq1d 7431	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → ((𝑎‘𝑏)(.r‘𝐿)𝑏) = ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))
106	105	mpteq2dva 5206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))
107	106	oveq2d 7432	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))
108	107	eqeq2d 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏))) ↔ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))))
109	92, 108	anbi12d 644	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → ((𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))))
110	109	adantlr 728	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → ((𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))))
111		fldextrspunlsp.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin)
112	111	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ∈ Fin)
113		ovexd 7451	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) ∈ V)
114		fvexd 6900	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (0g‘𝐿) ∈ V)
115	95, 112, 113, 114	fsuppmptdm 9339	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿))
116		fldextrspunlsplem.4	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))
117	116	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))
118	8	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐿 ∈ Ring)
119	118	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐿 ∈ Ring)
120	3	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
121	31	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐺 ⊆ (Base‘𝐿))
122	21	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑃:𝐻⟶𝐺)
123	122	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ 𝐺)
124	121, 123	sseldd 3939	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ (Base‘𝐿))
125	119	adantr 486	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ Ring)
126	68	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐹 ⊆ (Base‘𝐿))
127	3	ad4antr 745	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
128	58	ad4antr 745	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐹 ∈ (SubDRing‘𝐽))
129	11	ad4antr 745	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐻 ∈ (SubDRing‘𝐿))
130		ovexd 7451	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝐹 ↑m 𝐵) ∈ V)
131		simp-4r 796	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
132	129, 130, 131	elmaprd 33072	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
133		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ℎ ∈ 𝐻)
134	132, 133	ffvelcdmd 7084	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
135	127, 128, 134	elmaprd 33072	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑢‘ℎ):𝐵⟶𝐹)
136		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
137	135, 136	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
138	126, 137	sseldd 3939	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
139		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
140		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
141	139, 140	lbsss 21228	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
142	3, 141	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
143		eqidd 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
144	143, 61	srabase 21328	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
145	66, 144	eqtr2d 2801	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐽)‘𝐹)) = 𝐻)
146	142, 145	sseqtrd 3974	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
147	146, 63	sstrd 3948	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
148	147	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ⊆ (Base‘𝐿))
149	148	sselda 3938	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ (Base‘𝐿))
150	29, 19, 125, 138, 149	ringcld 20363	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
151		fvexd 6900	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (0g‘𝐿) ∈ V)
152		ssidd 3961	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ⊆ 𝐵)
153	58	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐽))
154	11	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐻 ∈ (SubDRing‘𝐿))
155		ovexd 7451	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐹 ↑m 𝐵) ∈ V)
156		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
157	154, 155, 156	elmaprd 33072	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
158	157	ffvelcdmda 7083	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
159	120, 153, 158	elmaprd 33072	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ):𝐵⟶𝐹)
160	52	breq1d 5121	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓) finSupp (0g‘𝐿) ↔ (𝑢‘ℎ) finSupp (0g‘𝐿)))
161		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
162	52	fveq1d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑏) = ((𝑢‘ℎ)‘𝑏))
163	162	oveq1d 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))
164	163	mpteq2dv 5207	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))
165	164	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))
166	161, 165	eqeq12d 2781	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))) ↔ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))))
167	160, 166	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑢‘ℎ) finSupp (0g‘𝐿) ∧ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))))
168		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
169		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ ∈ 𝐻)
170	167, 168, 169	rspcdva 3584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ) finSupp (0g‘𝐿) ∧ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))))
171	170	simpld 500	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) finSupp (0g‘𝐿))
172	119	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝐿 ∈ Ring)
173		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝑦 ∈ (Base‘𝐿))
174	29, 19, 5, 172, 173	ringlzd 20404	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → ((0g‘𝐿)(.r‘𝐿)𝑦) = (0g‘𝐿))
175	151, 151, 120, 152, 149, 159, 171, 174	fisuppov1 33075	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)) finSupp (0g‘𝐿))
176	29, 5, 19, 119, 120, 124, 150, 175	gsummulc2 20424	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))) = ((𝑃‘ℎ)(.r‘𝐿)(𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
177	124	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑃‘ℎ) ∈ (Base‘𝐿))
178	29, 19, 125, 177, 138, 149	ringassd 20364	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))
179	178	mpteq2dva 5206	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)) = (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))))
180	179	oveq2d 7432	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
181	170	simprd 501	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))
182		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → ((𝑢‘ℎ)‘𝑏) = ((𝑢‘ℎ)‘𝑐))
183		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → 𝑏 = 𝑐)
184	182, 183	oveq12d 7434	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
185	184	cbvmptv 5217	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
186	185	oveq2i 7427	. . . . . . . . . . . 12 ⊢ (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))
187	181, 186	eqtrdi 2816	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))))
188	187	oveq2d 7432	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)ℎ) = ((𝑃‘ℎ)(.r‘𝐿)(𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
189	176, 180, 188	3eqtr4rd 2811	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)ℎ) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))
190	189	mpteq2dva 5206	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ)) = (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)))))
191	190	oveq2d 7432	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
192	51, 161	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)𝑓) = ((𝑃‘ℎ)(.r‘𝐿)ℎ))
193	192	cbvmptv 5217	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))
194	193	oveq2i 7427	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ)))
195	194	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))))
196	9	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐿 ∈ CMnd)
197	8	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐿 ∈ Ring)
198	31	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐺 ⊆ (Base‘𝐿))
199	80	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ 𝐺)
200	198, 199	sseldd 3939	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ (Base‘𝐿))
201	29, 19, 197, 200, 79	ringcld 20363	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) ∈ (Base‘𝐿))
202	147	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ⊆ (Base‘𝐿))
203	202	sselda 3938	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ (Base‘𝐿))
204	203	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝑐 ∈ (Base‘𝐿))
205	29, 19, 197, 201, 204	ringcld 20363	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
206	205	anasss 472	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
207	81	fsuppimpd 9332	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ∈ Fin)
208	207	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑃 supp (0g‘𝐿)) ∈ Fin)
209		suppssdm 8175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 supp (0g‘𝐿)) ⊆ dom 𝑃
210	209, 21	fssdm 6729	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ⊆ 𝐻)
211	210	sseld 3937	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
212	211	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
213		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → (𝑢‘𝑓) finSupp (0g‘𝐿))
214	213	fsuppimpd 9332	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)
215	214	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → ((𝑢‘𝑓) finSupp (0g‘𝐿) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
216	215	adantrd 497	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
217	212, 216	imim12d 82	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → ((𝑓 ∈ 𝐻 → ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)))
218	217	ralimdv2 3176	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) → ∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
219	218	imp 412	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)
220		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑢‘𝑓) = (𝑢‘𝑖))
221	220	oveq1d 7431	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → ((𝑢‘𝑓) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
222	221	eleq1d 2850	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → (((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin))
223	222	cbvralvw 3245	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
224	219, 223	sylib 221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
225		iunfi 9303	. . . . . . . . . . 11 ⊢ (((𝑃 supp (0g‘𝐿)) ∈ Fin ∧ ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
226	208, 224, 225	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
227		xpfi 9282	. . . . . . . . . 10 ⊢ ((∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin ∧ (𝑃 supp (0g‘𝐿)) ∈ Fin) → (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))) ∈ Fin)
228	226, 208, 227	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))) ∈ Fin)
229		snssi 4753	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝑃 supp (0g‘𝐿)) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
230	229	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑃 supp (0g‘𝐿))) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
231	230	iunxpssiun1 32960	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ⊆ (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))))
232	231	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ⊆ (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))))
233	228, 232	ssfid 9232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ∈ Fin)
234	21	ffnd 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 Fn 𝐻)
235	234	ad6antr 749	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑃 Fn 𝐻)
236	11	ad6antr 749	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐻 ∈ (SubDRing‘𝐿))
237		fvexd 6900	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (0g‘𝐿) ∈ V)
238		simpllr 788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ℎ ∈ 𝐻)
239		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ¬ ℎ ∈ (𝑃 supp (0g‘𝐿)))
240	238, 239	eldifd 3917	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ℎ ∈ (𝐻 ∖ (𝑃 supp (0g‘𝐿))))
241	235, 236, 237, 240	fvdifsupp 8169	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑃‘ℎ) = (0g‘𝐿))
242	241	oveq1d 7431	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = ((0g‘𝐿)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
243	8	ad6antr 749	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐿 ∈ Ring)
244	68	ad6antr 749	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐹 ⊆ (Base‘𝐿))
245	3	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
246	58	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐹 ∈ (SubDRing‘𝐽))
247		ovexd 7451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝐹 ↑m 𝐵) ∈ V)
248		simp-6r 800	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
249	236, 247, 248	elmaprd 33072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
250	249, 238	ffvelcdmd 7084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
251	245, 246, 250	elmaprd 33072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑢‘ℎ):𝐵⟶𝐹)
252		simp-4r 796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑐 ∈ 𝐵)
253	251, 252	ffvelcdmd 7084	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
254	244, 253	sseldd 3939	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
255	29, 19, 5, 243, 254	ringlzd 20404	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((0g‘𝐿)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
256	242, 255	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
257	3	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
258	58	ad6antr 749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐹 ∈ (SubDRing‘𝐽))
259	11	ad6antr 749	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐻 ∈ (SubDRing‘𝐿))
260		ovexd 7451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝐹 ↑m 𝐵) ∈ V)
261		simp-6r 800	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
262	259, 260, 261	elmaprd 33072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
263		simpllr 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ℎ ∈ 𝐻)
264	262, 263	ffvelcdmd 7084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
265	257, 258, 264	elmaprd 33072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ):𝐵⟶𝐹)
266	265	ffnd 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ) Fn 𝐵)
267		fvexd 6900	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (0g‘𝐿) ∈ V)
268		simp-4r 796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑐 ∈ 𝐵)
269		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿)))
270	268, 269	eldifd 3917	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑐 ∈ (𝐵 ∖ ((𝑢‘ℎ) supp (0g‘𝐿))))
271	266, 257, 267, 270	fvdifsupp 8169	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) = (0g‘𝐿))
272	271	oveq2d 7432	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)(0g‘𝐿)))
273	197	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐿 ∈ Ring)
274	200	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑃‘ℎ) ∈ (Base‘𝐿))
275	29, 19, 5, 273, 274	ringrzd 20405	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)(0g‘𝐿)) = (0g‘𝐿))
276	272, 275	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
277		df-br 5112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ ⟨𝑐, ℎ⟩ ∈ ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
278		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑖 → (𝑢‘ℎ) = (𝑢‘𝑖))
279	278	oveq1d 7431	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → ((𝑢‘ℎ) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
280		sneq 4601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
281	279, 280	xpeq12d 5694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑖 → (((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = (((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
282	281	cbviunv 5005	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})
283	282	eleq2i 2857	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑐, ℎ⟩ ∈ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) ↔ ⟨𝑐, ℎ⟩ ∈ ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
284		opeliun2xp 5731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑐, ℎ⟩ ∈ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) ↔ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
285	277, 283, 284	3bitr2i 302	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
286	285	notbii 323	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ ¬ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
287		ianor 997	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) ↔ (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
288	286, 287	sylbb 222	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ → (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
289	288	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
290	256, 276, 289	mpjaodan 973	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
291	290	oveq1d 7431	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = ((0g‘𝐿)(.r‘𝐿)𝑐))
292	118	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → 𝐿 ∈ Ring)
293	203	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → 𝑐 ∈ (Base‘𝐿))
294	29, 19, 5, 292, 293	ringlzd 20404	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → ((0g‘𝐿)(.r‘𝐿)𝑐) = (0g‘𝐿))
295	291, 294	eqtrd 2800	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
296	295	an42ds 1520	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
297	296	an32s 665	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
298	297	anasss 472	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
299	298	an32s 665	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
300	299	anasss 472	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ((𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
301	29, 5, 196, 4, 154, 206, 233, 300	gsumcom3 20072	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
302	191, 195, 301	3eqtr4d 2810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
303	118	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ Ring)
304	29, 5, 19, 303, 12, 203, 201, 86	gsummulc1 20423	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))) = ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
305	304	mpteq2dva 5206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)))) = (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐)))
306	305	oveq2d 7432	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))))
307	117, 302, 306	3eqtrd 2804	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))))
308	51, 162	oveq12d 7434	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
309	308	cbvmptv 5217	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
310	182	oveq2d 7432	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
311	310	mpteq2dv 5207	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
312	309, 311	eqtrid 2812	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
313	312	oveq2d 7432	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))))
314	313, 183	oveq12d 7434	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏) = ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
315	314	cbvmptv 5217	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
316	315	oveq2i 7427	. . . . 5 ⊢ (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐)))
317	307, 316	eqtr4di 2818	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))
318	115, 317	jca 521	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))))
319	90, 110, 318	rspcedvd 3585	. 2 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))
320		breq1 5114	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒 finSupp (0g‘𝐿) ↔ (𝑢‘𝑓) finSupp (0g‘𝐿)))
321		fveq1 6884	. . . . . . . 8 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒‘𝑏) = ((𝑢‘𝑓)‘𝑏))
322	321	oveq1d 7431	. . . . . . 7 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))
323	322	mpteq2dv 5207	. . . . . 6 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))
324	323	oveq2d 7432	. . . . 5 ⊢ (𝑒 = (𝑢‘𝑓) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))
325	324	eqeq2d 2776	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))) ↔ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
326	320, 325	anbi12d 644	. . 3 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))))
327		ovexd 7451	. . 3 ⊢ (𝜑 → (𝐹 ↑m 𝐵) ∈ V)
328		eqid 2765	. . . . . . . . . 10 ⊢ (LSpan‘((subringAlg ‘𝐽)‘𝐹)) = (LSpan‘((subringAlg ‘𝐽)‘𝐹))
329	139, 140, 328	lbssp 21230	. . . . . . . . 9 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
330	3, 329	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
331	144, 66, 330	3eqtr4rd 2811	. . . . . . 7 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = 𝐻)
332	331	eleq2d 2851	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ 𝑓 ∈ 𝐻))
333		eqid 2765	. . . . . . 7 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
334		eqid 2765	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐽)‘𝐹)) = (Scalar‘((subringAlg ‘𝐽)‘𝐹))
335		eqid 2765	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
336		eqid 2765	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))
337		sdrgsubrg 20924	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ∈ (SubRing‘𝐽))
338	58, 337	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
339		eqid 2765	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
340	339	sralmod 21338	. . . . . . . 8 ⊢ (𝐹 ∈ (SubRing‘𝐽) → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
341	338, 340	syl 18	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
342	328, 139, 333, 334, 335, 336, 341, 142	ellspds 33723	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))))
343	332, 342	bitr3d 284	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ 𝐻 ↔ ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))))
344	343	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))))
345		eqid 2765	. . . . . . . . . 10 ⊢ (𝐽 ↾s 𝐹) = (𝐽 ↾s 𝐹)
346	345, 59	ressbas2 17316	. . . . . . . . 9 ⊢ (𝐹 ⊆ (Base‘𝐽) → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
347	61, 346	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
348	143, 61	srasca 21331	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐽)‘𝐹)))
349	348	fveq2d 6889	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐽 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
350	347, 349	eqtr2d 2801	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = 𝐹)
351	350	oveq1d 7431	. . . . . 6 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) = (𝐹 ↑m 𝐵))
352		sdrgsubrg 20924	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
353	11, 352	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
354		subrgsubg 20706	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ∈ (SubGrp‘𝐿))
355	64, 5	subg0 19222	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → (0g‘𝐿) = (0g‘𝐽))
356	353, 354, 355	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐽))
357	64	sdrgdrng 20923	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
358	11, 357	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ DivRing)
359	358	drngringd 20865	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Ring)
360	359	ringcmnd 20392	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ CMnd)
361	360	cmnmndd 19898	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Mnd)
362		subrgsubg 20706	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐽) → 𝐹 ∈ (SubGrp‘𝐽))
363		eqid 2765	. . . . . . . . . . . . 13 ⊢ (0g‘𝐽) = (0g‘𝐽)
364	363	subg0cl 19224	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubGrp‘𝐽) → (0g‘𝐽) ∈ 𝐹)
365	338, 362, 364	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐽) ∈ 𝐹)
366	345, 59, 363	ress0g 18842	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Mnd ∧ (0g‘𝐽) ∈ 𝐹 ∧ 𝐹 ⊆ (Base‘𝐽)) → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
367	361, 365, 61, 366	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
368	348	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(𝐽 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
369	356, 367, 368	3eqtrrd 2805	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘𝐿))
370	369	breq2d 5123	. . . . . . . 8 ⊢ (𝜑 → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
371	370	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
372	3	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
373		subgsubm 19239	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → 𝐻 ∈ (SubMnd‘𝐿))
374	353, 354, 373	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubMnd‘𝐿))
375	374	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐻 ∈ (SubMnd‘𝐿))
376	64, 19	ressmulr 17378	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → (.r‘𝐿) = (.r‘𝐽))
377	11, 376	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐽))
378	143, 61	sravsca 21332	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐽) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
379	377, 378	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
380	379	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
381	380	oveqd 7433	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))
382	353	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐻 ∈ (SubRing‘𝐿))
383	67	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐹 ⊆ 𝐻)
384	25	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐹 ∈ (SubDRing‘𝐼))
385	351	eleq2d 2851	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) ↔ 𝑒 ∈ (𝐹 ↑m 𝐵)))
386	385	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒 ∈ (𝐹 ↑m 𝐵))
387	372, 384, 386	elmaprd 33072	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒:𝐵⟶𝐹)
388	387	ffvelcdmda 7083	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐹)
389	383, 388	sseldd 3939	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐻)
390	146	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ⊆ 𝐻)
391	390	sselda 3938	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐻)
392	19, 382, 389, 391	subrgmcld 33591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) ∈ 𝐻)
393	381, 392	eqeltrrd 2866	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) ∈ 𝐻)
394	393	fmpttd 7114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)):𝐵⟶𝐻)
395	372, 375, 394, 64	gsumsubm 18918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
396	377, 378	eqtr2d 2801	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
397	396	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
398	397	oveqd 7433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) = ((𝑒‘𝑏)(.r‘𝐿)𝑏))
399	398	mpteq2dv 5207	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) = (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))
400	399	oveq2d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))
401	3	mptexd 7226	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) ∈ V)
402		fvexd 6900	. . . . . . . . . . 11 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ V)
403	339, 401, 358, 402, 61	gsumsra 33407	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
404	403	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
405	395, 400, 404	3eqtr3rd 2809	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))
406	405	eqeq2d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) ↔ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))))
407	371, 406	anbi12d 644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ((𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ (𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
408	351, 407	rexeqbidva 3332	. . . . 5 ⊢ (𝜑 → (∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
409	408	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → (∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
410	344, 409	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))))
411	326, 11, 327, 410	ac6mapd 33015	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
412	319, 411	r19.29a 3175	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  {csn 4591  ⟨cop 4597  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   supp csupp 8158   ↑_m cmap 8826  Fincfn 8945   finSupp cfsupp 9324  Basecbs 17287   ↾_s cress 17308  ._rcmulr 17329  Scalarcsca 17331   ·_𝑠 cvsca 17332  0_gc0g 17510   Σ_g cgsu 17511  Mndcmnd 18814  SubMndcsubmnd 18864  SubGrpcsubg 19210  CMndccmn 19874  Ringcrg 20339  SubRingcsubrg 20698  RingSpancrgspn 20739  DivRingcdr 20857  Fieldcfield 20858  SubDRingcsdrg 20919  LModclmod 21011  LSpanclspn 21122  LBasisclbs 21225  subringAlg csra 21322

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-reg 9557  ax-inf2 9613  ax-ac2 10458  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-sup 9405  df-oi 9475  df-r1 9739  df-rank 9740  df-scott 9861  df-card 9937  df-ac 10112  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-fzo 13696  df-seq 14052  df-hash 14381  df-struct 17225  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-mulr 17342  df-sca 17344  df-vsca 17345  df-ip 17346  df-tset 17347  df-ple 17348  df-ds 17350  df-hom 17352  df-cco 17353  df-0g 17512  df-gsum 17513  df-prds 17518  df-pws 17520  df-mre 17656  df-mrc 17657  df-acs 17659  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-mhm 18865  df-submnd 18866  df-grp 19027  df-minusg 19028  df-sbg 19029  df-mulg 19158  df-subg 19213  df-ghm 19308  df-cntz 19411  df-cmn 19876  df-abl 19877  df-mgp 20241  df-rng 20255  df-ur 20288  df-ring 20341  df-nzr 20640  df-subrng 20675  df-subrg 20699  df-drng 20859  df-field 20860  df-sdrg 20920  df-lmod 21013  df-lss 21083  df-lsp 21123  df-lmhm 21173  df-lbs 21226  df-sra 21324  df-rgmod 21325  df-dsmm 21912  df-frlm 21927  df-uvc 21963

This theorem is used by:  fldextrspunlsp  34104