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Theorem tsmsxplem2 24048

Description: Lemma for tsmsxp 24049. (Contributed by Mario Carneiro, 21-Sep-2015.)

**Hypotheses**
Ref	Expression
tsmsxp.b	⊢ 𝐵 = (Base‘𝐺)
tsmsxp.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmsxp.2	⊢ (𝜑 → 𝐺 ∈ TopGrp)
tsmsxp.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmsxp.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)
tsmsxp.f	⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
tsmsxp.h	⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
tsmsxp.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
tsmsxp.j	⊢ 𝐽 = (TopOpen‘𝐺)
tsmsxp.z	⊢ 0 = (0_g‘𝐺)
tsmsxp.p	⊢ + = (+_g‘𝐺)
tsmsxp.m	⊢ − = (-_g‘𝐺)
tsmsxp.l	⊢ (𝜑 → 𝐿 ∈ 𝐽)
tsmsxp.3	⊢ (𝜑 → 0 ∈ 𝐿)
tsmsxp.k	⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))
tsmsxp.4	⊢ (𝜑 → ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈)
tsmsxp.n	⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin))
tsmsxp.s	⊢ (𝜑 → 𝐷 ⊆ (𝐾 × 𝑁))
tsmsxp.x	⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿)
tsmsxp.5	⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆)
tsmsxp.6	⊢ (𝜑 → ∀𝑔 ∈ (𝐿 ↑_m 𝐾)(𝐺 Σ_g 𝑔) ∈ 𝑇)

**Assertion**
Ref	Expression
tsmsxplem2	⊢ (𝜑 → (𝐺 Σ_g (𝐻 ↾ 𝐾)) ∈ 𝑈)

Distinct variable groups: 𝑔,𝑘, 0 𝑐,𝑑,𝑔,𝑗,𝑘,𝑥,𝐺 𝐵,𝑔,𝑘 𝐷,𝑔,𝑗,𝑘,𝑥 𝑔,𝐿,𝑗,𝑥 𝐴,𝑔,𝑗,𝑘 𝐾,𝑐,𝑑,𝑔,𝑗,𝑘,𝑥 𝑆,𝑐 𝐻,𝑑,𝑔,𝑗,𝑘,𝑥 𝑁,𝑐,𝑑,𝑔,𝑥 𝑈,𝑐,𝑑 − ,𝑑,𝑔,𝑗,𝑥 𝐶,𝑔,𝑗,𝑘 𝑇,𝑐,𝑑,𝑔 + ,𝑐,𝑑,𝑔 𝐹,𝑐,𝑑,𝑔,𝑗,𝑘,𝑥 𝜑,𝑔,𝑗,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑐,𝑑) 𝐴(𝑥,𝑐,𝑑) 𝐵(𝑥,𝑗,𝑐,𝑑) 𝐶(𝑥,𝑐,𝑑) 𝐷(𝑐,𝑑) + (𝑥,𝑗,𝑘) 𝑆(𝑥,𝑔,𝑗,𝑘,𝑑) 𝑇(𝑥,𝑗,𝑘) 𝑈(𝑥,𝑔,𝑗,𝑘) 𝐻(𝑐) 𝐽(𝑥,𝑔,𝑗,𝑘,𝑐,𝑑) 𝐿(𝑘,𝑐,𝑑) − (𝑘,𝑐) 𝑁(𝑗,𝑘) 𝑉(𝑥,𝑔,𝑗,𝑘,𝑐,𝑑) 𝑊(𝑥,𝑔,𝑗,𝑘,𝑐,𝑑) 0 (𝑥,𝑗,𝑐,𝑑)

Proof of Theorem tsmsxplem2 Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsxp.2	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TopGrp)
2		tgpgrp 23972	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		tsmsxp.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd)
5		isabl 19721	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
6	3, 4, 5	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		tsmsxp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
8		tsmsxp.z	. . . 4 ⊢ 0 = (0_g‘𝐺)
9		tsmsxp.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))
10	9	elin2d 4171	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Fin)
11		tsmsxp.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin))
12	11	elin2d 4171	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
13		xpfi 9276	. . . . 5 ⊢ ((𝐾 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝐾 × 𝑁) ∈ Fin)
14	10, 12, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐾 × 𝑁) ∈ Fin)
15		tsmsxp.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
16		elfpw 9312	. . . . . . . 8 ⊢ (𝐾 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐾 ⊆ 𝐴 ∧ 𝐾 ∈ Fin))
17	16	simplbi 497	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾 ⊆ 𝐴)
18	9, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ 𝐴)
19		elfpw 9312	. . . . . . . 8 ⊢ (𝑁 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑁 ⊆ 𝐶 ∧ 𝑁 ∈ Fin))
20	19	simplbi 497	. . . . . . 7 ⊢ (𝑁 ∈ (𝒫 𝐶 ∩ Fin) → 𝑁 ⊆ 𝐶)
21	11, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝐶)
22		xpss12 5656	. . . . . 6 ⊢ ((𝐾 ⊆ 𝐴 ∧ 𝑁 ⊆ 𝐶) → (𝐾 × 𝑁) ⊆ (𝐴 × 𝐶))
23	18, 21, 22	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐾 × 𝑁) ⊆ (𝐴 × 𝐶))
24	15, 23	fssresd 6730	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐾 × 𝑁)):(𝐾 × 𝑁)⟶𝐵)
25		tsmsxp.3	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐿)
26	24, 14, 25	fdmfifsupp 9333	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐾 × 𝑁)) finSupp 0 )
27	7, 8, 4, 14, 24, 26	gsumcl 19852	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝐵)
28		tsmsxp.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
29	28, 18	fssresd 6730	. . . 4 ⊢ (𝜑 → (𝐻 ↾ 𝐾):𝐾⟶𝐵)
30	29, 10, 25	fdmfifsupp 9333	. . . 4 ⊢ (𝜑 → (𝐻 ↾ 𝐾) finSupp 0 )
31	7, 8, 4, 9, 29, 30	gsumcl 19852	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝐻 ↾ 𝐾)) ∈ 𝐵)
32		tsmsxp.p	. . . 4 ⊢ + = (+_g‘𝐺)
33		tsmsxp.m	. . . 4 ⊢ − = (-_g‘𝐺)
34	7, 32, 33	ablpncan3 19753	. . 3 ⊢ ((𝐺 ∈ Abel ∧ ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝐵 ∧ (𝐺 Σ_g (𝐻 ↾ 𝐾)) ∈ 𝐵)) → ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))) = (𝐺 Σ_g (𝐻 ↾ 𝐾)))
35	6, 27, 31, 34	syl12anc 836	. 2 ⊢ (𝜑 → ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))) = (𝐺 Σ_g (𝐻 ↾ 𝐾)))
36		tsmsxp.5	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆)
37	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝐺 ∈ CMnd)
38		snfi 9017	. . . . . . . . 9 ⊢ {𝑦} ∈ Fin
39	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑁 ∈ Fin)
40		xpfi 9276	. . . . . . . . 9 ⊢ (({𝑦} ∈ Fin ∧ 𝑁 ∈ Fin) → ({𝑦} × 𝑁) ∈ Fin)
41	38, 39, 40	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ({𝑦} × 𝑁) ∈ Fin)
42	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
43	18	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑦 ∈ 𝐴)
44	43	snssd 4776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → {𝑦} ⊆ 𝐴)
45	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑁 ⊆ 𝐶)
46		xpss12 5656	. . . . . . . . . 10 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑁 ⊆ 𝐶) → ({𝑦} × 𝑁) ⊆ (𝐴 × 𝐶))
47	44, 45, 46	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ({𝑦} × 𝑁) ⊆ (𝐴 × 𝐶))
48	42, 47	fssresd 6730	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹 ↾ ({𝑦} × 𝑁)):({𝑦} × 𝑁)⟶𝐵)
49	8	fvexi 6875	. . . . . . . . . 10 ⊢ 0 ∈ V
50	49	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 0 ∈ V)
51	48, 41, 50	fdmfifsupp 9333	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹 ↾ ({𝑦} × 𝑁)) finSupp 0 )
52	7, 8, 37, 41, 48, 51	gsumcl 19852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) ∈ 𝐵)
53	52	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))):𝐾⟶𝐵)
54		eqid 2730	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) = (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))
55		ovexd 7425	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) ∈ V)
56	54, 10, 55, 25	fsuppmptdm 9334	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) finSupp 0 )
57	7, 8, 33, 6, 9, 29, 53, 30, 56	gsumsub 19885	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g ((𝐻 ↾ 𝐾) ∘_f − (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) = ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))))
58		fvexd 6876	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐻‘𝑦) ∈ V)
59	28, 18	feqresmpt 6933	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ 𝐾) = (𝑦 ∈ 𝐾 ↦ (𝐻‘𝑦)))
60		eqidd 2731	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) = (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))
61	9, 58, 55, 59, 60	offval2 7676	. . . . . 6 ⊢ (𝜑 → ((𝐻 ↾ 𝐾) ∘_f − (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) = (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))))
62	61	oveq2d 7406	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g ((𝐻 ↾ 𝐾) ∘_f − (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) = (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))))
63		cmnmnd 19734	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
64	37, 63	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝐺 ∈ Mnd)
65		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑦 ∈ 𝐾)
66	42	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
67	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → 𝑦 ∈ 𝐴)
68	45	sselda 3949	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → 𝑧 ∈ 𝐶)
69	66, 67, 68	fovcdmd 7564	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦𝐹𝑧) ∈ 𝐵)
70	69	fmpttd 7090	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)):𝑁⟶𝐵)
71		eqid 2730	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))
72		ovexd 7425	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦𝐹𝑧) ∈ V)
73	71, 39, 72, 50	fsuppmptdm 9334	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)) finSupp 0 )
74	7, 8, 37, 39, 70, 73	gsumcl 19852	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))) ∈ 𝐵)
75		velsn 4608	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {𝑦} ↔ 𝑤 = 𝑦)
76		ovres 7558	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑦} ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑤𝐹𝑧))
77	75, 76	sylanbr 582	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑤𝐹𝑧))
78		oveq1 7397	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (𝑤𝐹𝑧) = (𝑦𝐹𝑧))
79	78	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤𝐹𝑧) = (𝑦𝐹𝑧))
80	77, 79	eqtrd 2765	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑦𝐹𝑧))
81	80	mpteq2dva 5203	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)))
82	81	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))))
83	7, 82	gsumsn 19891	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐾 ∧ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))) ∈ 𝐵) → (𝐺 Σ_g (𝑤 ∈ {𝑦} ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧))))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))))
84	64, 65, 74, 83	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝑤 ∈ {𝑦} ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧))))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))))
85	38	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → {𝑦} ∈ Fin)
86	7, 8, 37, 85, 39, 48, 51	gsumxp 19913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) = (𝐺 Σ_g (𝑤 ∈ {𝑦} ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧))))))
87		ovres 7558	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑁) → (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧) = (𝑦𝐹𝑧))
88	87	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧) = (𝑦𝐹𝑧))
89	88	mpteq2dva 5203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)))
90	89	oveq2d 7406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))))
91	84, 86, 90	3eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧))))
92	91	mpteq2dva 5203	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) = (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧)))))
93	92	oveq2d 7406	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) = (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧))))))
94	7, 8, 4, 10, 12, 24, 26	gsumxp 19913	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) = (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧))))))
95	93, 94	eqtr4d 2768	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))
96	95	oveq2d 7406	. . . . 5 ⊢ (𝜑 → ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) = ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))))
97	57, 62, 96	3eqtr3d 2773	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) = ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))))
98		oveq2 7398	. . . . . 6 ⊢ (𝑔 = (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) → (𝐺 Σ_g 𝑔) = (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))))
99	98	eleq1d 2814	. . . . 5 ⊢ (𝑔 = (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) → ((𝐺 Σ_g 𝑔) ∈ 𝑇 ↔ (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) ∈ 𝑇))
100		tsmsxp.6	. . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ (𝐿 ↑_m 𝐾)(𝐺 Σ_g 𝑔) ∈ 𝑇)
101		tsmsxp.x	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿)
102		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐻‘𝑥) = (𝐻‘𝑦))
103		sneq 4602	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
104	103	xpeq1d 5670	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝑁) = ({𝑦} × 𝑁))
105	104	reseq2d 5953	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ ({𝑥} × 𝑁)) = (𝐹 ↾ ({𝑦} × 𝑁)))
106	105	oveq2d 7406	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁))) = (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))
107	102, 106	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) = ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))
108	107	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿 ↔ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) ∈ 𝐿))
109	108	rspccva 3590	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿 ∧ 𝑦 ∈ 𝐾) → ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) ∈ 𝐿)
110	101, 109	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) ∈ 𝐿)
111	110	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))):𝐾⟶𝐿)
112		tsmsxp.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ 𝐽)
113	112, 9	elmapd 8816	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) ∈ (𝐿 ↑_m 𝐾) ↔ (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))):𝐾⟶𝐿))
114	111, 113	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) ∈ (𝐿 ↑_m 𝐾))
115	99, 100, 114	rspcdva 3592	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) ∈ 𝑇)
116	97, 115	eqeltrrd 2830	. . 3 ⊢ (𝜑 → ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))) ∈ 𝑇)
117		tsmsxp.4	. . 3 ⊢ (𝜑 → ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈)
118		oveq1 7397	. . . . 5 ⊢ (𝑐 = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) → (𝑐 + 𝑑) = ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑))
119	118	eleq1d 2814	. . . 4 ⊢ (𝑐 = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) → ((𝑐 + 𝑑) ∈ 𝑈 ↔ ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑) ∈ 𝑈))
120		oveq2 7398	. . . . 5 ⊢ (𝑑 = ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))) → ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑) = ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))))
121	120	eleq1d 2814	. . . 4 ⊢ (𝑑 = ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))) → (((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑) ∈ 𝑈 ↔ ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))) ∈ 𝑈))
122	119, 121	rspc2va 3603	. . 3 ⊢ ((((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆 ∧ ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁)))) ∈ 𝑇) ∧ ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈) → ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))) ∈ 𝑈)
123	36, 116, 117, 122	syl21anc 837	. 2 ⊢ (𝜑 → ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + ((𝐺 Σ_g (𝐻 ↾ 𝐾)) − (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))) ∈ 𝑈)
124	35, 123	eqeltrrd 2830	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐻 ↾ 𝐾)) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 𝒫 cpw 4566 {csn 4592 ↦ cmpt 5191 × cxp 5639 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ↑_m cmap 8802 Fincfn 8921 Basecbs 17186 +_gcplusg 17227 TopOpenctopn 17391 0_gc0g 17409 Σ_g cgsu 17410 Mndcmnd 18668 Grpcgrp 18872 -_gcsg 18874 CMndccmn 19717 Abelcabl 19718 TopGrpctgp 23965 tsums ctsu 24020

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

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-0g 17411 df-gsum 17412 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-tgp 23967

This theorem is referenced by: tsmsxp 24049

W3C validator