Theorem fldextrspunlsplem 33817

Description: Lemma for fldextrspunlsp 33818: 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 727	. . . 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 727	. . . 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 2736	. . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿)
6		fldextrspunfld.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Field)
7	6	flddrngd 20718	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
8	7	drngringd 20714	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
9	8	ringcmnd 20265	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ CMnd)
10	9	ad3antrrr 731	. . . . . 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 731	. . . . . 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 20768	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
15		subrgsubg 20554	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
16		subgsubm 19124	. . . . . . . 8 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → 𝐺 ∈ (SubMnd‘𝐿))
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubMnd‘𝐿))
18	17	ad3antrrr 731	. . . . . 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 2736	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
20	14	ad3antrrr 731	. . . . . . . . 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 731	. . . . . . . . . 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 7037	. . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐼) = (Base‘𝐼)
27	26	sdrgss 20770	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ⊆ (Base‘𝐼))
28	25, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐼))
29		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐿) = (Base‘𝐿)
30	29	sdrgss 20770	. . . . . . . . . . . . . 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 17208	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
34	31, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
35	28, 34	sseqtrrd 3959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐺)
36	35	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
37	3	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
38	25	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐼))
39	11	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐻 ∈ (SubDRing‘𝐿))
40		ovexd 7402	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝐹 ↑m 𝐵) ∈ V)
41		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
42	39, 40, 41	elmaprd 32753	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
43	42, 23	ffvelcdmd 7037	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓) ∈ (𝐹 ↑m 𝐵))
44	37, 38, 43	elmaprd 32753	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓):𝐵⟶𝐹)
45		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑐 ∈ 𝐵)
46	44, 45	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐹)
47	36, 46	sseldd 3922	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐺)
48	19, 20, 24, 47	subrgmcld 33293	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) ∈ 𝐺)
49	48	fmpttd 7067	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
50	49	adantlr 716	. . . . . 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 6840	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑃‘𝑓) = (𝑃‘ℎ))
52		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑢‘𝑓) = (𝑢‘ℎ))
53	52	fveq1d 6842	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘ℎ)‘𝑐))
54	51, 53	oveq12d 7385	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
55	54	cbvmptv 5189	. . . . . . 7 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
56		fvexd 6855	. . . . . . . 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 3945	. . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
60	59	sdrgss 20770	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
61	58, 60	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
62	29	sdrgss 20770	. . . . . . . . . . . . . 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 17208	. . . . . . . . . . . . 13 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
66	63, 65	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
67	61, 66	sseqtrrd 3959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
68	67, 63	sstrd 3932	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
69	68	ad4antr 733	. . . . . . . . 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 733	. . . . . . . . . . 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 733	. . . . . . . . . . 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 7402	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 32753	. . . . . . . . . . . 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 7036	. . . . . . . . . . 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 32753	. . . . . . . . . 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 769	. . . . . . . . . 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 7037	. . . . . . . . 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 3922	. . . . . . . 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 731	. . . . . . . 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 731	. . . . . . . 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 733	. . . . . . . . 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 20276	. . . . . . . 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 32756	. . . . . . 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 5120	. . . . . 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 19894	. . . . 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 7067	. . . 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 8788	. . 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 5088	. . . . . 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 769	. . . . . . . . . . 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 6842	. . . . . . . . . 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 2736	. . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑏 → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘𝑓)‘𝑏))
97	96	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))
98	97	mpteq2dv 5179	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))
99	98	oveq2d 7383	. . . . . . . . . . . 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 7402	. . . . . . . . . . . 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 6971	. . . . . . . . . . 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 716	. . . . . . . . . 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 2771	. . . . . . . . 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 7382	. . . . . . . 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 5178	. . . . . . 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 7383	. . . . . 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 2747	. . . . 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 633	. . . 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 716	. . 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 727	. . . . 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 7402	. . . . 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 6855	. . . . 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 9289	. . . 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 727	. . . . . 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 727	. . . . . . . . . . . 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 731	. . . . . . . . . . 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 731	. . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 7036	. . . . . . . . . . . 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 3922	. . . . . . . . . . 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 733	. . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . 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 3922	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
140		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
141	139, 140	lbsss 21072	. . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
144	143, 61	srabase 21172	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
145	66, 144	eqtr2d 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐽)‘𝐹)) = 𝐻)
146	142, 145	sseqtrd 3958	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
147	146, 63	sstrd 3932	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
148	147	ad3antrrr 731	. . . . . . . . . . . . 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 3921	. . . . . . . . . . . 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 20241	. . . . . . . . . . 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 6855	. . . . . . . . . . . 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 3945	. . . . . . . . . . . 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 731	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . 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 32753	. . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓) finSupp (0g‘𝐿) ↔ (𝑢‘ℎ) finSupp (0g‘𝐿)))
161		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
162	52	fveq1d 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑏) = ((𝑢‘ℎ)‘𝑏))
163	162	oveq1d 7382	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))
164	163	mpteq2dv 5179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))
165	164	oveq2d 7383	. . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . 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 20276	. . . . . . . . . . . 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 32756	. . . . . . . . . . 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 20296	. . . . . . . . . 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 20238	. . . . . . . . . . . 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 5178	. . . . . . . . . . 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 7383	. . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → ((𝑢‘ℎ)‘𝑏) = ((𝑢‘ℎ)‘𝑐))
183		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → 𝑏 = 𝑐)
184	182, 183	oveq12d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
185	184	cbvmptv 5189	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
186	185	oveq2i 7378	. . . . . . . . . . . 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 2787	. . . . . . . . . . 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 7383	. . . . . . . . . 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 2782	. . . . . . . . 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 5178	. . . . . . . 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 7383	. . . . . . 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 7385	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)𝑓) = ((𝑃‘ℎ)(.r‘𝐿)ℎ))
193	192	cbvmptv 5189	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))
194	193	oveq2i 7378	. . . . . . . 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 727	. . . . . . . 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 733	. . . . . . . . . 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 733	. . . . . . . . . . . 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 7036	. . . . . . . . . . . 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 3922	. . . . . . . . . . 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 20241	. . . . . . . . . 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 727	. . . . . . . . . . . 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 3921	. . . . . . . . . . 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 20241	. . . . . . . . 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 9282	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ∈ Fin)
208	207	ad2antrr 727	. . . . . . . . . . 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 8127	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 supp (0g‘𝐿)) ⊆ dom 𝑃
210	209, 21	fssdm 6687	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ⊆ 𝐻)
211	210	sseld 3920	. . . . . . . . . . . . . . . 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 9282	. . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑢‘𝑓) = (𝑢‘𝑖))
221	220	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → ((𝑢‘𝑓) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
222	221	eleq1d 2821	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → (((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin))
223	222	cbvralvw 3215	. . . . . . . . . . . 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 9253	. . . . . . . . . . 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 585	. . . . . . . . . 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 9230	. . . . . . . . . 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 585	. . . . . . . . 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 4729	. . . . . . . . . . . 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 32638	. . . . . . . . . 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 727	. . . . . . . . 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 9179	. . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 Fn 𝐻)
235	234	ad6antr 737	. . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . 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 8121	. . . . . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . 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 20276	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . 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 3900	. . . . . . . . . . . . . . . . . . 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 8121	. . . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . 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 20277	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 5086	. . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑖 → (𝑢‘ℎ) = (𝑢‘𝑖))
279	278	oveq1d 7382	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → ((𝑢‘ℎ) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
280		sneq 4577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
281	279, 280	xpeq12d 5662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑖 → (((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = (((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
282	281	cbviunv 4981	. . . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . 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 5699	. . . . . . . . . . . . . . . . . . . 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 984	. . . . . . . . . . . . . . . . . 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 961	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 20276	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 1492	. . . . . . . . . . . 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 653	. . . . . . . . . . 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 653	. . . . . . . . 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 19953	. . . . . . 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 2781	. . . . . 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 20295	. . . . . . . 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 5178	. . . . . . 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 7383	. . . . . 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 2775	. . . . 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 7385	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
309	308	cbvmptv 5189	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
310	182	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
311	310	mpteq2dv 5179	. . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
313	312	oveq2d 7383	. . . . . . . 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 7385	. . . . . . 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 5189	. . . . . 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 7378	. . . . 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 2789	. . . 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 3566	. 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 5088	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒 finSupp (0g‘𝐿) ↔ (𝑢‘𝑓) finSupp (0g‘𝐿)))
321		fveq1 6839	. . . . . . . 8 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒‘𝑏) = ((𝑢‘𝑓)‘𝑏))
322	321	oveq1d 7382	. . . . . . 7 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))
323	322	mpteq2dv 5179	. . . . . 6 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))
324	323	oveq2d 7383	. . . . 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 2747	. . . 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 633	. . 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 7402	. . 3 ⊢ (𝜑 → (𝐹 ↑m 𝐵) ∈ V)
328		eqid 2736	. . . . . . . . . 10 ⊢ (LSpan‘((subringAlg ‘𝐽)‘𝐹)) = (LSpan‘((subringAlg ‘𝐽)‘𝐹))
329	139, 140, 328	lbssp 21074	. . . . . . . . 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 2782	. . . . . . 7 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = 𝐻)
332	331	eleq2d 2822	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ 𝑓 ∈ 𝐻))
333		eqid 2736	. . . . . . 7 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
334		eqid 2736	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐽)‘𝐹)) = (Scalar‘((subringAlg ‘𝐽)‘𝐹))
335		eqid 2736	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
336		eqid 2736	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))
337		sdrgsubrg 20768	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ∈ (SubRing‘𝐽))
338	58, 337	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
339		eqid 2736	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
340	339	sralmod 21182	. . . . . . . 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 33428	. . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝐽 ↾s 𝐹) = (𝐽 ↾s 𝐹)
346	345, 59	ressbas2 17208	. . . . . . . . 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 21175	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐽)‘𝐹)))
349	348	fveq2d 6844	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐽 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
350	347, 349	eqtr2d 2772	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = 𝐹)
351	350	oveq1d 7382	. . . . . 6 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) = (𝐹 ↑m 𝐵))
352		sdrgsubrg 20768	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
353	11, 352	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
354		subrgsubg 20554	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ∈ (SubGrp‘𝐿))
355	64, 5	subg0 19108	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → (0g‘𝐿) = (0g‘𝐽))
356	353, 354, 355	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐽))
357	64	sdrgdrng 20767	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
358	11, 357	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ DivRing)
359	358	drngringd 20714	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Ring)
360	359	ringcmnd 20265	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ CMnd)
361	360	cmnmndd 19779	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Mnd)
362		subrgsubg 20554	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐽) → 𝐹 ∈ (SubGrp‘𝐽))
363		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0g‘𝐽) = (0g‘𝐽)
364	363	subg0cl 19110	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubGrp‘𝐽) → (0g‘𝐽) ∈ 𝐹)
365	338, 362, 364	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐽) ∈ 𝐹)
366	345, 59, 363	ress0g 18730	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Mnd ∧ (0g‘𝐽) ∈ 𝐹 ∧ 𝐹 ⊆ (Base‘𝐽)) → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
367	361, 365, 61, 366	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
368	348	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(𝐽 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
369	356, 367, 368	3eqtrrd 2776	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘𝐿))
370	369	breq2d 5097	. . . . . . . 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 19124	. . . . . . . . . . . 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 17270	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → (.r‘𝐿) = (.r‘𝐽))
377	11, 376	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐽))
378	143, 61	sravsca 21176	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐽) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
379	377, 378	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
380	379	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
381	380	oveqd 7384	. . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐻 ∈ (SubRing‘𝐿))
383	67	ad2antrr 727	. . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . 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 32753	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒:𝐵⟶𝐹)
388	387	ffvelcdmda 7036	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐹)
389	383, 388	sseldd 3922	. . . . . . . . . . . . 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 3921	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐻)
392	19, 382, 389, 391	subrgmcld 33293	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) ∈ 𝐻)
393	381, 392	eqeltrrd 2837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) ∈ 𝐻)
394	393	fmpttd 7067	. . . . . . . . . 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 18803	. . . . . . . . 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 2772	. . . . . . . . . . . . 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 7384	. . . . . . . . . . 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 5179	. . . . . . . . . 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 7383	. . . . . . . . 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 7179	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) ∈ V)
402		fvexd 6855	. . . . . . . . . . 11 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ V)
403	339, 401, 358, 402, 61	gsumsra 33108	. . . . . . . . . 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 2780	. . . . . . . 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 2747	. . . . . . 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 633	. . . . . 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 3302	. . . . 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 32696	. 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 3145	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 848   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  {csn 4567  ⟨cop 4573  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893   finSupp cfsupp 9274  Basecbs 17179   ↾_s cress 17200  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18702  SubMndcsubmnd 18750  SubGrpcsubg 19096  CMndccmn 19755  Ringcrg 20214  SubRingcsubrg 20546  RingSpancrgspn 20587  DivRingcdr 20706  Fieldcfield 20707  SubDRingcsdrg 20763  LModclmod 20855  LSpanclspn 20966  LBasisclbs 21069  subringAlg csra 21166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-r1 9688  df-rank 9689  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-nzr 20490  df-subrng 20523  df-subrg 20547  df-drng 20708  df-field 20709  df-sdrg 20764  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-lbs 21070  df-sra 21168  df-rgmod 21169  df-dsmm 21712  df-frlm 21727  df-uvc 21763

This theorem is referenced by:  fldextrspunlsp  33818