Theorem fldextrspunlsplem 33675

Description: Lemma for fldextrspunlsp 33676: 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 726	. . . 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 726	. . . 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 2730	. . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿)
6		fldextrspunfld.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Field)
7	6	flddrngd 20657	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
8	7	drngringd 20653	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
9	8	ringcmnd 20200	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ CMnd)
10	9	ad3antrrr 730	. . . . . 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 730	. . . . . 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 20707	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
15		subrgsubg 20493	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
16		subgsubm 19087	. . . . . . . 8 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → 𝐺 ∈ (SubMnd‘𝐿))
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubMnd‘𝐿))
18	17	ad3antrrr 730	. . . . . 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 2730	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
20	14	ad3antrrr 730	. . . . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑃:𝐻⟶𝐺)
23		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ 𝐻)
24	22, 23	ffvelcdmd 7060	. . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐼) = (Base‘𝐼)
27	26	sdrgss 20709	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ⊆ (Base‘𝐼))
28	25, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐼))
29		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐿) = (Base‘𝐿)
30	29	sdrgss 20709	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
31	1, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
32		fldextrspunfld.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (𝐿 ↾s 𝐺)
33	32, 29	ressbas2 17215	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
34	31, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
35	28, 34	sseqtrrd 3987	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐺)
36	35	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
37	3	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
38	25	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐼))
39	11	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐻 ∈ (SubDRing‘𝐿))
40		ovexd 7425	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝐹 ↑m 𝐵) ∈ V)
41		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
42	39, 40, 41	elmaprd 32610	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
43	42, 23	ffvelcdmd 7060	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓) ∈ (𝐹 ↑m 𝐵))
44	37, 38, 43	elmaprd 32610	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓):𝐵⟶𝐹)
45		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑐 ∈ 𝐵)
46	44, 45	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐹)
47	36, 46	sseldd 3950	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐺)
48	19, 20, 24, 47	subrgmcld 33191	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) ∈ 𝐺)
49	48	fmpttd 7090	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
50	49	adantlr 715	. . . . . 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 6861	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑃‘𝑓) = (𝑃‘ℎ))
52		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑢‘𝑓) = (𝑢‘ℎ))
53	52	fveq1d 6863	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘ℎ)‘𝑐))
54	51, 53	oveq12d 7408	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
55	54	cbvmptv 5214	. . . . . . 7 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
56		fvexd 6876	. . . . . . . 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 3973	. . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
60	59	sdrgss 20709	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
61	58, 60	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
62	29	sdrgss 20709	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
63	11, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
64		fldextrspunfld.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (𝐿 ↾s 𝐻)
65	64, 29	ressbas2 17215	. . . . . . . . . . . . 13 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
66	63, 65	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
67	61, 66	sseqtrrd 3987	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
68	67, 63	sstrd 3960	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
69	68	ad4antr 732	. . . . . . . . 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 732	. . . . . . . . . . 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 732	. . . . . . . . . . 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 7425	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 32610	. . . . . . . . . . . 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 7059	. . . . . . . . . . 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 32610	. . . . . . . . . 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 768	. . . . . . . . . 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 7060	. . . . . . . . 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 3950	. . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 732	. . . . . . . . 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 484	. . . . . . . . 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 20211	. . . . . . . 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 32613	. . . . . . 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 5145	. . . . . 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 19856	. . . . 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 7090	. . . 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 8817	. . 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 5113	. . . . . 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 481	. . . . 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 768	. . . . . . . . . . 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 6863	. . . . . . . . . 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 2730	. . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑏 → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘𝑓)‘𝑏))
97	96	oveq2d 7406	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))
98	97	mpteq2dv 5204	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))
99	98	oveq2d 7406	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
101		ovexd 7425	. . . . . . . . . . . 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 6994	. . . . . . . . . . 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 715	. . . . . . . . . 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 2765	. . . . . . . . 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 7405	. . . . . . . 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 5203	. . . . . . 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 7406	. . . . . 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 2741	. . . . 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 632	. . . 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 715	. . 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 726	. . . . 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 7425	. . . . 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 6876	. . . . 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 9334	. . . 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 726	. . . . . 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 726	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 7059	. . . . . . . . . . . 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 3950	. . . . . . . . . . 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 480	. . . . . . . . . . . 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 732	. . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . 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 3950	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
140		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
141	139, 140	lbsss 20991	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
142	3, 141	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
143		eqidd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
144	143, 61	srabase 21091	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
145	66, 144	eqtr2d 2766	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐽)‘𝐹)) = 𝐻)
146	142, 145	sseqtrd 3986	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
147	146, 63	sstrd 3960	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
148	147	ad3antrrr 730	. . . . . . . . . . . . 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 3949	. . . . . . . . . . . 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 20176	. . . . . . . . . . 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 6876	. . . . . . . . . . . 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 3973	. . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . 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 7059	. . . . . . . . . . . . 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 32610	. . . . . . . . . . . 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 5120	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓) finSupp (0g‘𝐿) ↔ (𝑢‘ℎ) finSupp (0g‘𝐿)))
161		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
162	52	fveq1d 6863	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑏) = ((𝑢‘ℎ)‘𝑏))
163	162	oveq1d 7405	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))
164	163	mpteq2dv 5204	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))
165	164	oveq2d 7406	. . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 3592	. . . . . . . . . . . . 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 494	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 20211	. . . . . . . . . . . 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 32613	. . . . . . . . . . 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 20233	. . . . . . . . . 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 480	. . . . . . . . . . . . 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 20173	. . . . . . . . . . . 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 5203	. . . . . . . . . . 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 7406	. . . . . . . . . 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 495	. . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → ((𝑢‘ℎ)‘𝑏) = ((𝑢‘ℎ)‘𝑐))
183		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → 𝑏 = 𝑐)
184	182, 183	oveq12d 7408	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
185	184	cbvmptv 5214	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
186	185	oveq2i 7401	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 7406	. . . . . . . . . 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 2776	. . . . . . . . 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 5203	. . . . . . . 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 7406	. . . . . . 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 7408	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)𝑓) = ((𝑃‘ℎ)(.r‘𝐿)ℎ))
193	192	cbvmptv 5214	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))
194	193	oveq2i 7401	. . . . . . . 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 726	. . . . . . . 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 732	. . . . . . . . . 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 732	. . . . . . . . . . . 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 7059	. . . . . . . . . . . 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 3950	. . . . . . . . . . 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 20176	. . . . . . . . . 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 726	. . . . . . . . . . . 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 3949	. . . . . . . . . . 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 480	. . . . . . . . . 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 20176	. . . . . . . . 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 466	. . . . . . . 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 9327	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ∈ Fin)
208	207	ad2antrr 726	. . . . . . . . . . 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 8159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 supp (0g‘𝐿)) ⊆ dom 𝑃
210	209, 21	fssdm 6710	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ⊆ 𝐻)
211	210	sseld 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
212	211	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
213		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → (𝑢‘𝑓) finSupp (0g‘𝐿))
214	213	fsuppimpd 9327	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)
215	214	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → ((𝑢‘𝑓) finSupp (0g‘𝐿) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
216	215	adantrd 491	. . . . . . . . . . . . . . 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 81	. . . . . . . . . . . . . 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 3143	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑢‘𝑓) = (𝑢‘𝑖))
221	220	oveq1d 7405	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → ((𝑢‘𝑓) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
222	221	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → (((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin))
223	222	cbvralvw 3216	. . . . . . . . . . . 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 218	. . . . . . . . . . 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 9301	. . . . . . . . . . 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 584	. . . . . . . . . 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 9276	. . . . . . . . . 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 584	. . . . . . . . 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 4775	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝑃 supp (0g‘𝐿)) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
230	229	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑃 supp (0g‘𝐿))) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
231	230	iunxpssiun1 32504	. . . . . . . . . 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 726	. . . . . . . . 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 9219	. . . . . . . 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 6692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 Fn 𝐻)
235	234	ad6antr 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . . . . 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 8153	. . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . . . . 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 20211	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . . . . . . . . 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 32610	. . . . . . . . . . . . . . . . . . . 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 6692	. . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . . . . 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 8153	. . . . . . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 20212	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑖 → (𝑢‘ℎ) = (𝑢‘𝑖))
279	278	oveq1d 7405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → ((𝑢‘ℎ) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
280		sneq 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
281	279, 280	xpeq12d 5672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑖 → (((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = (((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
282	281	cbviunv 5007	. . . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . 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 5709	. . . . . . . . . . . . . . . . . . . 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 299	. . . . . . . . . . . . . . . . . . 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 320	. . . . . . . . . . . . . . . . . 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 983	. . . . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 960	. . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 20211	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 32386	. . . . . . . . . . . 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 652	. . . . . . . . . . 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 466	. . . . . . . . . 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 652	. . . . . . . . 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 466	. . . . . . . 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 19915	. . . . . . 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 2775	. . . . . 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 480	. . . . . . . . 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 20232	. . . . . . . 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 5203	. . . . . . 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 7406	. . . . . 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 2769	. . . . 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 7408	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
309	308	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
310	182	oveq2d 7406	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
311	310	mpteq2dv 5204	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
313	312	oveq2d 7406	. . . . . . . 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 7408	. . . . . . 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 5214	. . . . . 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 7401	. . . . 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 2783	. . . 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 511	. . 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 3593	. 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 5113	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒 finSupp (0g‘𝐿) ↔ (𝑢‘𝑓) finSupp (0g‘𝐿)))
321		fveq1 6860	. . . . . . . 8 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒‘𝑏) = ((𝑢‘𝑓)‘𝑏))
322	321	oveq1d 7405	. . . . . . 7 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))
323	322	mpteq2dv 5204	. . . . . 6 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))
324	323	oveq2d 7406	. . . . 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 2741	. . . 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 632	. . 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 7425	. . 3 ⊢ (𝜑 → (𝐹 ↑m 𝐵) ∈ V)
328		eqid 2730	. . . . . . . . . 10 ⊢ (LSpan‘((subringAlg ‘𝐽)‘𝐹)) = (LSpan‘((subringAlg ‘𝐽)‘𝐹))
329	139, 140, 328	lbssp 20993	. . . . . . . . 9 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
330	3, 329	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
331	144, 66, 330	3eqtr4rd 2776	. . . . . . 7 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = 𝐻)
332	331	eleq2d 2815	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ 𝑓 ∈ 𝐻))
333		eqid 2730	. . . . . . 7 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
334		eqid 2730	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐽)‘𝐹)) = (Scalar‘((subringAlg ‘𝐽)‘𝐹))
335		eqid 2730	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
336		eqid 2730	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))
337		sdrgsubrg 20707	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ∈ (SubRing‘𝐽))
338	58, 337	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
339		eqid 2730	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
340	339	sralmod 21101	. . . . . . . 8 ⊢ (𝐹 ∈ (SubRing‘𝐽) → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
341	338, 340	syl 17	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
342	328, 139, 333, 334, 335, 336, 341, 142	ellspds 33346	. . . . . 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 281	. . . . 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 476	. . . 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 2730	. . . . . . . . . 10 ⊢ (𝐽 ↾s 𝐹) = (𝐽 ↾s 𝐹)
346	345, 59	ressbas2 17215	. . . . . . . . 9 ⊢ (𝐹 ⊆ (Base‘𝐽) → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
347	61, 346	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
348	143, 61	srasca 21094	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐽)‘𝐹)))
349	348	fveq2d 6865	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐽 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
350	347, 349	eqtr2d 2766	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = 𝐹)
351	350	oveq1d 7405	. . . . . 6 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) = (𝐹 ↑m 𝐵))
352		sdrgsubrg 20707	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
353	11, 352	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
354		subrgsubg 20493	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ∈ (SubGrp‘𝐿))
355	64, 5	subg0 19071	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → (0g‘𝐿) = (0g‘𝐽))
356	353, 354, 355	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐽))
357	64	sdrgdrng 20706	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
358	11, 357	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ DivRing)
359	358	drngringd 20653	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Ring)
360	359	ringcmnd 20200	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ CMnd)
361	360	cmnmndd 19741	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Mnd)
362		subrgsubg 20493	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐽) → 𝐹 ∈ (SubGrp‘𝐽))
363		eqid 2730	. . . . . . . . . . . . 13 ⊢ (0g‘𝐽) = (0g‘𝐽)
364	363	subg0cl 19073	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubGrp‘𝐽) → (0g‘𝐽) ∈ 𝐹)
365	338, 362, 364	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐽) ∈ 𝐹)
366	345, 59, 363	ress0g 18696	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Mnd ∧ (0g‘𝐽) ∈ 𝐹 ∧ 𝐹 ⊆ (Base‘𝐽)) → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
367	361, 365, 61, 366	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
368	348	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(𝐽 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
369	356, 367, 368	3eqtrrd 2770	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘𝐿))
370	369	breq2d 5122	. . . . . . . 8 ⊢ (𝜑 → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
371	370	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
372	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
373		subgsubm 19087	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → 𝐻 ∈ (SubMnd‘𝐿))
374	353, 354, 373	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubMnd‘𝐿))
375	374	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐻 ∈ (SubMnd‘𝐿))
376	64, 19	ressmulr 17277	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → (.r‘𝐿) = (.r‘𝐽))
377	11, 376	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐽))
378	143, 61	sravsca 21095	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐽) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
379	377, 378	eqtrd 2765	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
380	379	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
381	380	oveqd 7407	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐻 ∈ (SubRing‘𝐿))
383	67	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐹 ⊆ 𝐻)
384	25	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐹 ∈ (SubDRing‘𝐼))
385	351	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) ↔ 𝑒 ∈ (𝐹 ↑m 𝐵)))
386	385	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒 ∈ (𝐹 ↑m 𝐵))
387	372, 384, 386	elmaprd 32610	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒:𝐵⟶𝐹)
388	387	ffvelcdmda 7059	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐹)
389	383, 388	sseldd 3950	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐻)
390	146	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ⊆ 𝐻)
391	390	sselda 3949	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐻)
392	19, 382, 389, 391	subrgmcld 33191	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) ∈ 𝐻)
393	381, 392	eqeltrrd 2830	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) ∈ 𝐻)
394	393	fmpttd 7090	. . . . . . . . . 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 18769	. . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
397	396	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
398	397	oveqd 7407	. . . . . . . . . . 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 5204	. . . . . . . . . 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 7406	. . . . . . . . 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 7201	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) ∈ V)
402		fvexd 6876	. . . . . . . . . . 11 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ V)
403	339, 401, 358, 402, 61	gsumsra 32994	. . . . . . . . . 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 480	. . . . . . . . 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 2774	. . . . . . . 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 2741	. . . . . . 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 632	. . . . . 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 3308	. . . . 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 480	. . . 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 232	. . 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 32556	. 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 3142	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191   × cxp 5639   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   supp csupp 8142   ↑_m cmap 8802  Fincfn 8921   finSupp cfsupp 9319  Basecbs 17186   ↾_s cress 17207  ._rcmulr 17228  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  SubMndcsubmnd 18716  SubGrpcsubg 19059  CMndccmn 19717  Ringcrg 20149  SubRingcsubrg 20485  RingSpancrgspn 20526  DivRingcdr 20645  Fieldcfield 20646  SubDRingcsdrg 20702  LModclmod 20773  LSpanclspn 20884  LBasisclbs 20988  subringAlg csra 21085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601  ax-ac2 10423  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-r1 9724  df-rank 9725  df-card 9899  df-ac 10076  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-nzr 20429  df-subrng 20462  df-subrg 20486  df-drng 20647  df-field 20648  df-sdrg 20703  df-lmod 20775  df-lss 20845  df-lsp 20885  df-lmhm 20936  df-lbs 20989  df-sra 21087  df-rgmod 21088  df-dsmm 21648  df-frlm 21663  df-uvc 21699

This theorem is referenced by:  fldextrspunlsp  33676