Theorem gsumvsca1 33854

Description: Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)

Hypotheses

Ref	Expression
gsumvsca.b	⊢ 𝐵 = (Base‘𝑊)
gsumvsca.g	⊢ 𝐺 = (Scalar‘𝑊)
gsumvsca.z	⊢ 0 = (0g‘𝑊)
gsumvsca.t	⊢ · = ( ·𝑠 ‘𝑊)
gsumvsca.p	⊢ + = (+g‘𝑊)
gsumvsca.k	⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))
gsumvsca.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumvsca.w	⊢ (𝜑 → 𝑊 ∈ SLMod)
gsumvsca1.n	⊢ (𝜑 → 𝑃 ∈ 𝐾)
gsumvsca1.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵)

Assertion

Ref	Expression
gsumvsca1	⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))

Distinct variable groups:   · ,𝑘   𝐴,𝑘   𝑘,𝑊   𝜑,𝑘   𝐵,𝑘   𝑃,𝑘

Allowed substitution hints:   + (𝑘)   𝑄(𝑘)   𝐺(𝑘)   𝐾(𝑘)   0 (𝑘)

Proof of Theorem gsumvsca1

Dummy variables 𝑒 𝑎 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumvsca.a	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		ssid 3935	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3938	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 642	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		mpteq1 5197	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)))
6	5	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))))
7		mpteq1 5197	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ ∅ ↦ 𝑄))
8	7	oveq2d 7456	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)))
9	8	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
10	6, 9	eqeq12d 2779	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)))))
11	4, 10	imbi12d 347	. . . . 5 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))))
12		sseq1 3938	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
13	12	anbi2d 642	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
14		mpteq1 5197	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄)))
15	14	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))))
16		mpteq1 5197	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄))
17	16	oveq2d 7456	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))
18	17	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))
19	15, 18	eqeq12d 2779	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))))
20	13, 19	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝑒 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))))
21		sseq1 3938	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑧}) ⊆ 𝐴))
22	21	anbi2d 642	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)))
23		mpteq1 5197	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄)))
24	23	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))))
25		mpteq1 5197	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))
26	25	oveq2d 7456	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)))
27	26	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))
28	24, 27	eqeq12d 2779	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)))))
29	22, 28	imbi12d 347	. . . . 5 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
30		sseq1 3938	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
31	30	anbi2d 642	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
32		mpteq1 5197	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄)))
33	32	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))))
34		mpteq1 5197	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝐴 ↦ 𝑄))
35	34	oveq2d 7456	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))
36	35	oveq2d 7456	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
37	33, 36	eqeq12d 2779	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))))
38	31, 37	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))))
39		gsumvsca.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ SLMod)
40		gsumvsca.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))
41		gsumvsca1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐾)
42	40, 41	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐺))
43		gsumvsca.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑊)
44		gsumvsca.t	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑊)
45		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
46		gsumvsca.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
47	43, 44, 45, 46	slmdvs0 33853	. . . . . . . . 9 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺)) → (𝑃 · 0 ) = 0 )
48	39, 42, 47	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑃 · 0 ) = 0 )
49	48	eqcomd 2769	. . . . . . 7 ⊢ (𝜑 → 0 = (𝑃 · 0 ))
50		mpt0 6700	. . . . . . . . 9 ⊢ (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)) = ∅
51	50	oveq2i 7451	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑊 Σg ∅)
52	46	gsum0 18899	. . . . . . . 8 ⊢ (𝑊 Σg ∅) = 0
53	51, 52	eqtri 2786	. . . . . . 7 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = 0
54		mpt0 6700	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∅ ↦ 𝑄) = ∅
55	54	oveq2i 7451	. . . . . . . . 9 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = (𝑊 Σg ∅)
56	55, 52	eqtri 2786	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = 0
57	56	oveq2i 7451	. . . . . . 7 ⊢ (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))) = (𝑃 · 0 )
58	49, 53, 57	3eqtr4g 2823	. . . . . 6 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
59	58	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
60		ssun1 4179	. . . . . . . . 9 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑧})
61		sstr2 3920	. . . . . . . . 9 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑧}) → ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
62	60, 61	ax-mp 5	. . . . . . . 8 ⊢ ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
63	62	anim2i 629	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝜑 ∧ 𝑒 ⊆ 𝐴))
64	63	imim1i 64	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))))
65	39	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ SLMod)
66	42	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑃 ∈ (Base‘𝐺))
67		gsumvsca.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
68		slmdcmn 33833	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
69	65, 68	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ CMnd)
70		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ V)
72		simplrl 789	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝜑)
73		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑒 ∪ {𝑧}) ⊆ 𝐴)
74	73	unssad 4194	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ⊆ 𝐴)
75	74	sselda 3913	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑘 ∈ 𝐴)
76		gsumvsca1.c	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵)
77	72, 75, 76	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑄 ∈ 𝐵)
78	77	fmpttd 7136	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄):𝑒⟶𝐵)
79		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑒 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄)
80		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ Fin)
81	46	fvexi 6919	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
82	81	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 0 ∈ V)
83	79, 80, 77, 82	fsuppmptdm 9392	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄) finSupp 0 )
84	67, 46, 69, 71, 78, 83	gsumcl 20156	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) ∈ 𝐵)
85	73	unssbd 4195	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
86		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
87	86	snss 4745	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
88	85, 87	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
89	76	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
90	89	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
91		rspcsbela 4399	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵) → ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)
92	88, 90, 91	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)
93		gsumvsca.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
94	67, 93, 43, 44, 45	slmdvsdi 33843	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ SLMod ∧ (𝑃 ∈ (Base‘𝐺) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
95	65, 66, 84, 92, 94	syl13anc 1400	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
96	95	adantr 486	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
97		nfcsb1v 3875	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝑄
98	86	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
99		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑒)
100		csbeq1a 3865	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → 𝑄 = ⦋𝑧 / 𝑘⦌𝑄)
101	97, 67, 93, 69, 80, 77, 98, 99, 92, 100	gsumunsnf 20200	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄))
102	101	oveq2d 7456	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))) = (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)))
103	102	adantr 486	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))) = (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)))
104		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝑃
105		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
106	104, 105, 97	nfov 7470	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑃 · ⦋𝑧 / 𝑘⦌𝑄)
107	72, 39	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑊 ∈ SLMod)
108	72, 42	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑃 ∈ (Base‘𝐺))
109	67, 43, 44, 45	slmdvscl 33842	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵) → (𝑃 · 𝑄) ∈ 𝐵)
110	107, 108, 77, 109	syl3anc 1399	. . . . . . . . . . . 12 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → (𝑃 · 𝑄) ∈ 𝐵)
111	67, 43, 44, 45	slmdvscl 33842	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
112	65, 66, 92, 111	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
113	100	oveq2d 7456	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑃 · 𝑄) = (𝑃 · ⦋𝑧 / 𝑘⦌𝑄))
114	106, 67, 93, 69, 80, 110, 98, 99, 112, 113	gsumunsnf 20200	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
115	114	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
116		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))
117	116	oveq1d 7455	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
118	115, 117	eqtrd 2798	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
119	96, 103, 118	3eqtr4rd 2809	. . . . . . . 8 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))
120	119	exp31 425	. . . . . . 7 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
121	120	a2d 30	. . . . . 6 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
122	64, 121	syl5 35	. . . . 5 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
123	11, 20, 29, 38, 59, 122	findcard2s 9206	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))))
124	123	imp 412	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝜑 ∧ 𝐴 ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
125	2, 124	mpanr2 717	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
126	1, 125	mpancom 701	1 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588   ↦ cmpt 5189  ‘cfv 6555  (class class class)co 7440  Fincfn 8998  Basecbs 17413  +_gcplusg 17454  Scalarcsca 17457   ·_𝑠 cvsca 17458  0_gc0g 17636   Σ_g cgsu 17637  CMndccmn 20021  SLModcslmd 33828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813  df-seq 14169  df-hash 14499  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-0g 17638  df-gsum 17639  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-submnd 19005  df-mulg 19304  df-cntz 19558  df-cmn 20023  df-srg 20440  df-slmd 33829

This theorem is used by: (None)