Theorem fldextrspunlsp 33709

Description: Lemma for fldextrspunfld 33711. The subring generated by the union of two field extensions 𝐺 and 𝐻 is the vector sub- 𝐺 space generated by a basis 𝐵 of 𝐻. 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)

Assertion

Ref	Expression
fldextrspunlsp	⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

Proof of Theorem fldextrspunlsp

Dummy variables 𝑎 𝑓 𝑔 ℎ 𝑝 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextrspunlsp.c	. . . . 5 ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)))
3	2	eleq2d 2826	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻))))
4		eqid 2736	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
5		eqid 2736	. . . 4 ⊢ (.r‘𝐿) = (.r‘𝐿)
6		eqid 2736	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
7		fldextrspunlsp.n	. . . 4 ⊢ 𝑁 = (RingSpan‘𝐿)
8		fldextrspunfld.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
9	8	fldcrngd 20734	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
11		sdrgsubrg 20784	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
13		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
14		sdrgsubrg 20784	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
16	4, 5, 6, 7, 9, 12, 15	elrgspnsubrun 33241	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻)) ↔ ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))))
17	4	subrgss 20564	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
18	12, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
19		eqid 2736	. . . . . . . . 9 ⊢ (𝐿 ↾s 𝐺) = (𝐿 ↾s 𝐺)
20	19, 4	ressbas2 17279	. . . . . . . 8 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
21	18, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
22		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
23	22, 18	srasca 21175	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↾s 𝐺) = (Scalar‘((subringAlg ‘𝐿)‘𝐺)))
24	23	fveq2d 6908	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐿 ↾s 𝐺)) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
25	21, 24	eqtr2d 2777	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = 𝐺)
26	25	oveq1d 7444	. . . . 5 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵) = (𝐺 ↑m 𝐵))
27	9	crngringd 20239	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Ring)
28	27	ringcmnd 20273	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ CMnd)
29	28	cmnmndd 19818	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Mnd)
30		subrgsubg 20569	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
31	12, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝐿))
32	6	subg0cl 19148	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → (0g‘𝐿) ∈ 𝐺)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐿) ∈ 𝐺)
34	19, 4, 6	ress0g 18771	. . . . . . . . 9 ⊢ ((𝐿 ∈ Mnd ∧ (0g‘𝐿) ∈ 𝐺 ∧ 𝐺 ⊆ (Base‘𝐿)) → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
35	29, 33, 18, 34	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
36	23	fveq2d 6908	. . . . . . . 8 ⊢ (𝜑 → (0g‘(𝐿 ↾s 𝐺)) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
37	35, 36	eqtr2d 2777	. . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘𝐿))
38	37	breq2d 5153	. . . . . 6 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↔ 𝑎 finSupp (0g‘𝐿)))
39		eqid 2736	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
40		fldextrspunlsp.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
41	40	mptexd 7242	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) ∈ V)
42	39	sralmod 21186	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubRing‘𝐿) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
43	12, 42	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
44	39, 41, 8, 43, 18	gsumsra 33035	. . . . . . . 8 ⊢ (𝜑 → (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
45	22, 18	sravsca 21177	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)))
46	45	oveqd 7446	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))
47	46	mpteq2dv 5242	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) = (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))
48	47	oveq2d 7445	. . . . . . . 8 ⊢ (𝜑 → (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))))
49	44, 48	eqtr2d 2777	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))) = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
50	49	eqeq2d 2747	. . . . . 6 ⊢ (𝜑 → (𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))) ↔ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
51	38, 50	anbi12d 632	. . . . 5 ⊢ (𝜑 → ((𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))) ↔ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
52	26, 51	rexeqbidv 3346	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵)(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))) ↔ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
53		eqid 2736	. . . . 5 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
54		eqid 2736	. . . . 5 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
55		eqid 2736	. . . . 5 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
56		eqid 2736	. . . . 5 ⊢ (Scalar‘((subringAlg ‘𝐿)‘𝐺)) = (Scalar‘((subringAlg ‘𝐿)‘𝐺))
57		eqid 2736	. . . . 5 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
58		eqid 2736	. . . . 5 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))
59		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
60		eqid 2736	. . . . . . . . . 10 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
61	59, 60	lbsss 21068	. . . . . . . . 9 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
62	40, 61	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
63	4	subrgss 20564	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
64	15, 63	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
65		fldextrspunfld.j	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐿 ↾s 𝐻)
66	65, 4	ressbas2 17279	. . . . . . . . . 10 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
67	64, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
68		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
69		fldextrspunfld.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
70		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
71	70	sdrgss 20786	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
72	69, 71	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
73	68, 72	srabase 21169	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
74	67, 73	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
75	62, 74	sseqtrrd 4020	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
76	75, 64	sstrd 3993	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
77	22, 18	srabase 21169	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
78	76, 77	sseqtrd 4019	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
79	53, 54, 55, 56, 57, 58, 43, 78	ellspds 33383	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵)(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))))))
80		fldextrspunfld.k	. . . . . . 7 ⊢ 𝐾 = (𝐿 ↾s 𝐹)
81		fldextrspunfld.i	. . . . . . 7 ⊢ 𝐼 = (𝐿 ↾s 𝐺)
82	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐿 ∈ Field)
83		fldextrspunfld.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
84	83	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐹 ∈ (SubDRing‘𝐼))
85	69	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐹 ∈ (SubDRing‘𝐽))
86	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐺 ∈ (SubDRing‘𝐿))
87	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐻 ∈ (SubDRing‘𝐿))
88		fldextrspunlsp.e	. . . . . . 7 ⊢ 𝐸 = (𝐿 ↾s 𝐶)
89	40	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
90		fldextrspunlsp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
91	90	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐵 ∈ Fin)
92		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝 ∈ (𝐺 ↑m 𝐻))
93	87, 86, 92	elmaprd 32678	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝:𝐻⟶𝐺)
94		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝 finSupp (0g‘𝐿))
95		simprr 773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))
96		fveq2 6904	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑝‘𝑓) = (𝑝‘ℎ))
97		id 22	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
98	96, 97	oveq12d 7447	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = ((𝑝‘ℎ)(.r‘𝐿)ℎ))
99	98	cbvmptv 5253	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))
100	99	oveq2i 7440	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ)))
101	95, 100	eqtrdi 2792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑥 = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))))
102	80, 81, 65, 82, 84, 85, 86, 87, 7, 1, 88, 89, 91, 93, 94, 101	fldextrspunlsplem 33708	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
103	102	r19.29an 3157	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
104		breq1 5144	. . . . . . . 8 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑝 finSupp (0g‘𝐿) ↔ (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿)))
105		fveq1 6903	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑝‘𝑓) = ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓))
106	105	oveq1d 7444	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))
107	106	mpteq2dv 5242	. . . . . . . . . 10 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))
108	107	oveq2d 7445	. . . . . . . . 9 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
109	108	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) ↔ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))))
110	104, 109	anbi12d 632	. . . . . . 7 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → ((𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))))
111	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝐺 ∈ (SubDRing‘𝐿))
112	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝐻 ∈ (SubDRing‘𝐿))
113	40	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
114	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝐺 ∈ (SubDRing‘𝐿))
115		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 ∈ (𝐺 ↑m 𝐵))
116	113, 114, 115	elmaprd 32678	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎:𝐵⟶𝐺)
117	116	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) → 𝑎:𝐵⟶𝐺)
118	117	ffvelcdmda 7102	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
119	33	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ 𝐺)
120	118, 119	ifclda 4559	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ 𝐺)
121	120	fmpttd 7133	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))):𝐻⟶𝐺)
122	111, 112, 121	elmapdd 8877	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) ∈ (𝐺 ↑m 𝐻))
123		fvexd 6919	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (0g‘𝐿) ∈ V)
124	121	ffund 6738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → Fun (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))))
125		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝑎 finSupp (0g‘𝐿))
126	116	ffnd 6735	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 Fn 𝐵)
127	126	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑎 Fn 𝐵)
128	40	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
129		fvexd 6919	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ V)
130		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
131		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
132	131	eldifbd 3963	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → ¬ 𝑔 ∈ (𝑎 supp (0g‘𝐿)))
133	130, 132	eldifd 3961	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
134	127, 128, 129, 133	fvdifsupp 8192	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) = (0g‘𝐿))
135		eqidd 2737	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) = (0g‘𝐿))
136	134, 135	ifeqda 4560	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (0g‘𝐿))
137	136, 112	suppss2 8221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) supp (0g‘𝐿)) ⊆ (𝑎 supp (0g‘𝐿)))
138	122, 123, 124, 125, 137	fsuppsssuppgd 9418	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿))
139		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))
140		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 = 𝑓)
141		suppssdm 8198	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 supp (0g‘𝐿)) ⊆ dom 𝑎
142	116	fdmd 6744	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → dom 𝑎 = 𝐵)
143	142	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → dom 𝑎 = 𝐵)
144	141, 143	sseqtrid 4025	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐵)
145	144	sselda 3982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐵)
146	145	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑓 ∈ 𝐵)
147	140, 146	eqeltrd 2840	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 ∈ 𝐵)
148	147	iftrued 4532	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (𝑎‘𝑔))
149		fveq2 6904	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → (𝑎‘𝑔) = (𝑎‘𝑓))
150	149	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → (𝑎‘𝑔) = (𝑎‘𝑓))
151	148, 150	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (𝑎‘𝑓))
152	75	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ 𝐻)
153	144, 152	sstrd 3993	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐻)
154	153	sselda 3982	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐻)
155		fvexd 6919	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (𝑎‘𝑓) ∈ V)
156	139, 151, 154, 155	fvmptd2 7022	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) = (𝑎‘𝑓))
157	156	oveq1d 7444	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑓)(.r‘𝐿)𝑓))
158	157	mpteq2dva 5240	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)) = (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑓)(.r‘𝐿)𝑓)))
159		fveq2 6904	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑎‘𝑓) = (𝑎‘𝑣))
160		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → 𝑓 = 𝑣)
161	159, 160	oveq12d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑎‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑣)(.r‘𝐿)𝑣))
162	161	cbvmptv 5253	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑓)(.r‘𝐿)𝑓)) = (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))
163	158, 162	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)) = (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))
164	163	oveq2d 7445	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
165	28	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐿 ∈ CMnd)
166	13	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ∈ (SubDRing‘𝐿))
167		eleq1w 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∈ 𝐵 ↔ 𝑓 ∈ 𝐵))
168	167, 149	ifbieq1d 4548	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)))
169		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
170	169	eldifad 3962	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ 𝐻)
171		fvexd 6919	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑓) ∈ V)
172		fvexd 6919	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
173	171, 172	ifcld 4570	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)) ∈ V)
174	139, 168, 170, 173	fvmptd3 7037	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) = if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)))
175	174	oveq1d 7444	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = (if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿))(.r‘𝐿)𝑓))
176	126	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑎 Fn 𝐵)
177	40	ad4antr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
178		fvexd 6919	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (0g‘𝐿) ∈ V)
179		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
180		simplr 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
181	180	eldifbd 3963	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → ¬ 𝑓 ∈ (𝑎 supp (0g‘𝐿)))
182	179, 181	eldifd 3961	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
183	176, 177, 178, 182	fvdifsupp 8192	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (𝑎‘𝑓) = (0g‘𝐿))
184		eqidd 2737	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ ¬ 𝑓 ∈ 𝐵) → (0g‘𝐿) = (0g‘𝐿))
185	183, 184	ifeqda 4560	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)) = (0g‘𝐿))
186	185	oveq1d 7444	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿))(.r‘𝐿)𝑓) = ((0g‘𝐿)(.r‘𝐿)𝑓))
187	27	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
188	166, 14, 63	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ⊆ (Base‘𝐿))
189	188	ssdifssd 4146	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐻 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
190	189	sselda 3982	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (Base‘𝐿))
191	4, 5, 6, 187, 190	ringlzd 20284	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑓) = (0g‘𝐿))
192	175, 186, 191	3eqtrd 2780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = (0g‘𝐿))
193		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎 finSupp (0g‘𝐿))
194	193	fsuppimpd 9405	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ∈ Fin)
195	27	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝐿 ∈ Ring)
196	18	ad4antr 732	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
197	116	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → 𝑎:𝐵⟶𝐺)
198	197	ffvelcdmda 7102	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
199	196, 198	sseldd 3983	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ (Base‘𝐿))
200	18, 33	sseldd 3983	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0g‘𝐿) ∈ (Base‘𝐿))
201	200	ad4antr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ (Base‘𝐿))
202	199, 201	ifclda 4559	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ (Base‘𝐿))
203	202	fmpttd 7133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))):𝐻⟶(Base‘𝐿))
204	203	ffvelcdmda 7102	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) ∈ (Base‘𝐿))
205	188	sselda 3982	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ (Base‘𝐿))
206	4, 5, 195, 204, 205	ringcld 20252	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) ∈ (Base‘𝐿))
207	4, 6, 165, 166, 192, 194, 206, 153	gsummptres2 33041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
208	113	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
209	126	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑎 Fn 𝐵)
210	208	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
211		fvexd 6919	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
212		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
213	209, 210, 211, 212	fvdifsupp 8192	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑣) = (0g‘𝐿))
214	213	oveq1d 7444	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((0g‘𝐿)(.r‘𝐿)𝑣))
215	27	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
216	76	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ (Base‘𝐿))
217	216	ssdifssd 4146	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐵 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
218	217	sselda 3982	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (Base‘𝐿))
219	4, 5, 6, 215, 218	ringlzd 20284	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑣) = (0g‘𝐿))
220	214, 219	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = (0g‘𝐿))
221	27	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐿 ∈ Ring)
222	18	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
223	116	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎:𝐵⟶𝐺)
224	223	ffvelcdmda 7102	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ 𝐺)
225	222, 224	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ (Base‘𝐿))
226	216	sselda 3982	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ (Base‘𝐿))
227	4, 5, 221, 225, 226	ringcld 20252	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) ∈ (Base‘𝐿))
228	4, 6, 165, 208, 220, 194, 227, 144	gsummptres2 33041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (𝐿 Σg (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
229	164, 207, 228	3eqtr4d 2786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
230	229	eqeq2d 2747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) ↔ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
231	230	biimpar 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
232	231	anasss 466	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
233	138, 232	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))))
234	110, 122, 233	rspcedvdw 3624	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))))
235	234	r19.29an 3157	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))))
236	103, 235	impbida 801	. . . 4 ⊢ (𝜑 → (∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
237	52, 79, 236	3bitr4rd 312	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ 𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)))
238	3, 16, 237	3bitrd 305	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)))
239	238	eqrdv 2734	1 ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947   ∪ cun 3948   ⊆ wss 3950  ifcif 4524   class class class wbr 5141   ↦ cmpt 5223  dom cdm 5683   Fn wfn 6554  ⟶wf 6555  ‘cfv 6559  (class class class)co 7429   supp csupp 8181   ↑_m cmap 8862  Fincfn 8981   finSupp cfsupp 9397  Basecbs 17243   ↾_s cress 17270  ._rcmulr 17294  Scalarcsca 17296   ·_𝑠 cvsca 17297  0_gc0g 17480   Σ_g cgsu 17481  Mndcmnd 18743  SubGrpcsubg 19134  CMndccmn 19794  Ringcrg 20226  SubRingcsubrg 20561  RingSpancrgspn 20602  Fieldcfield 20722  SubDRingcsdrg 20779  LModclmod 20850  LSpanclspn 20961  LBasisclbs 21065  subringAlg csra 21162

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-reg 9628  ax-inf2 9677  ax-ac2 10499  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228  ax-pre-sup 11229  ax-addf 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-iin 4992  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-se 5636  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-isom 6568  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-of 7694  df-om 7884  df-1st 8010  df-2nd 8011  df-supp 8182  df-tpos 8247  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-2o 8503  df-er 8741  df-map 8864  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fsupp 9398  df-sup 9478  df-oi 9546  df-r1 9800  df-rank 9801  df-card 9975  df-ac 10152  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-div 11917  df-nn 12263  df-2 12325  df-3 12326  df-4 12327  df-5 12328  df-6 12329  df-7 12330  df-8 12331  df-9 12332  df-n0 12523  df-xnn0 12596  df-z 12610  df-dec 12730  df-uz 12875  df-rp 13031  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-word 14549  df-lsw 14597  df-concat 14605  df-s1 14630  df-substr 14675  df-pfx 14705  df-s2 14883  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17244  df-ress 17271  df-plusg 17306  df-mulr 17307  df-starv 17308  df-sca 17309  df-vsca 17310  df-ip 17311  df-tset 17312  df-ple 17313  df-ds 17315  df-unif 17316  df-hom 17317  df-cco 17318  df-0g 17482  df-gsum 17483  df-prds 17488  df-pws 17490  df-mre 17625  df-mrc 17626  df-acs 17628  df-mgm 18649  df-sgrp 18728  df-mnd 18744  df-mhm 18792  df-submnd 18793  df-grp 18950  df-minusg 18951  df-sbg 18952  df-mulg 19082  df-subg 19137  df-ghm 19227  df-cntz 19331  df-cmn 19796  df-abl 19797  df-mgp 20134  df-rng 20146  df-ur 20175  df-ring 20228  df-cring 20229  df-oppr 20326  df-nzr 20505  df-subrng 20538  df-subrg 20562  df-rgspn 20603  df-drng 20723  df-field 20724  df-sdrg 20780  df-lmod 20852  df-lss 20922  df-lsp 20962  df-lmhm 21013  df-lbs 21066  df-sra 21164  df-rgmod 21165  df-cnfld 21357  df-zring 21450  df-dsmm 21744  df-frlm 21759  df-uvc 21795  df-ind 32823

This theorem is referenced by:  fldextrspunlem1  33710