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Theorem tsmsxplem2 24183

Description: Lemma for tsmsxp 24184. (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 24107	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		tsmsxp.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd)
5		isabl 19826	. . . 4 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
6	3, 4, 5	sylanbrc 582	. . 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 4228	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Fin)
11		tsmsxp.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin))
12	11	elin2d 4228	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
13		xpfi 9386	. . . . 5 ⊢ ((𝐾 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝐾 × 𝑁) ∈ Fin)
14	10, 12, 13	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐾 × 𝑁) ∈ Fin)
15		tsmsxp.f	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
16		elfpw 9424	. . . . . . . 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 9424	. . . . . . . 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 5715	. . . . . 6 ⊢ ((𝐾 ⊆ 𝐴 ∧ 𝑁 ⊆ 𝐶) → (𝐾 × 𝑁) ⊆ (𝐴 × 𝐶))
23	18, 21, 22	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐾 × 𝑁) ⊆ (𝐴 × 𝐶))
24	15, 23	fssresd 6788	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐾 × 𝑁)):(𝐾 × 𝑁)⟶𝐵)
25		tsmsxp.3	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐿)
26	24, 14, 25	fdmfifsupp 9444	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐾 × 𝑁)) finSupp 0 )
27	7, 8, 4, 14, 24, 26	gsumcl 19957	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝐵)
28		tsmsxp.h	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
29	28, 18	fssresd 6788	. . . 4 ⊢ (𝜑 → (𝐻 ↾ 𝐾):𝐾⟶𝐵)
30	29, 10, 25	fdmfifsupp 9444	. . . 4 ⊢ (𝜑 → (𝐻 ↾ 𝐾) finSupp 0 )
31	7, 8, 4, 9, 29, 30	gsumcl 19957	. . 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 19858	. . 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 9109	. . . . . . . . 9 ⊢ {𝑦} ∈ Fin
39	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑁 ∈ Fin)
40		xpfi 9386	. . . . . . . . 9 ⊢ (({𝑦} ∈ Fin ∧ 𝑁 ∈ Fin) → ({𝑦} × 𝑁) ∈ Fin)
41	38, 39, 40	sylancr 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ({𝑦} × 𝑁) ∈ Fin)
42	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
43	18	sselda 4008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑦 ∈ 𝐴)
44	43	snssd 4834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → {𝑦} ⊆ 𝐴)
45	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 𝑁 ⊆ 𝐶)
46		xpss12 5715	. . . . . . . . . 10 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑁 ⊆ 𝐶) → ({𝑦} × 𝑁) ⊆ (𝐴 × 𝐶))
47	44, 45, 46	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ({𝑦} × 𝑁) ⊆ (𝐴 × 𝐶))
48	42, 47	fssresd 6788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹 ↾ ({𝑦} × 𝑁)):({𝑦} × 𝑁)⟶𝐵)
49	8	fvexi 6934	. . . . . . . . . 10 ⊢ 0 ∈ V
50	49	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → 0 ∈ V)
51	48, 41, 50	fdmfifsupp 9444	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐹 ↾ ({𝑦} × 𝑁)) finSupp 0 )
52	7, 8, 37, 41, 48, 51	gsumcl 19957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) ∈ 𝐵)
53	52	fmpttd 7149	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))):𝐾⟶𝐵)
54		eqid 2740	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) = (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))
55		ovexd 7483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) ∈ V)
56	54, 10, 55, 25	fsuppmptdm 9445	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) finSupp 0 )
57	7, 8, 33, 6, 9, 29, 53, 30, 56	gsumsub 19990	. . . . 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 6935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐻‘𝑦) ∈ V)
59	28, 18	feqresmpt 6991	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ 𝐾) = (𝑦 ∈ 𝐾 ↦ (𝐻‘𝑦)))
60		eqidd 2741	. . . . . . 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 7734	. . . . . 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 7464	. . . . 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 19839	. . . . . . . . . . . 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 4008	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → 𝑧 ∈ 𝐶)
69	66, 67, 68	fovcdmd 7622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦𝐹𝑧) ∈ 𝐵)
70	69	fmpttd 7149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)):𝑁⟶𝐵)
71		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))
72		ovexd 7483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦𝐹𝑧) ∈ V)
73	71, 39, 72, 50	fsuppmptdm 9445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)) finSupp 0 )
74	7, 8, 37, 39, 70, 73	gsumcl 19957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))) ∈ 𝐵)
75		velsn 4664	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {𝑦} ↔ 𝑤 = 𝑦)
76		ovres 7616	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑦} ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑤𝐹𝑧))
77	75, 76	sylanbr 581	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑤𝐹𝑧))
78		oveq1 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (𝑤𝐹𝑧) = (𝑦𝐹𝑧))
79	78	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤𝐹𝑧) = (𝑦𝐹𝑧))
80	77, 79	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = 𝑦 ∧ 𝑧 ∈ 𝑁) → (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧) = (𝑦𝐹𝑧))
81	80	mpteq2dva 5266	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)))
82	81	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑤(𝐹 ↾ ({𝑦} × 𝑁))𝑧))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧))))
83	7, 82	gsumsn 19996	. . . . . . . . . . 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 1371	. . . . . . . . . 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 20018	. . . . . . . . . 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 7616	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑁) → (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧) = (𝑦𝐹𝑧))
88	87	adantll 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑁) → (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧) = (𝑦𝐹𝑧))
89	88	mpteq2dva 5266	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧)) = (𝑧 ∈ 𝑁 ↦ (𝑦𝐹𝑧)))
90	89	oveq2d 7464	. . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))) = (𝐺 Σ_g (𝑧 ∈ 𝑁 ↦ (𝑦(𝐹 ↾ (𝐾 × 𝑁))𝑧))))
92	91	mpteq2dva 5266	. . . . . . . 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 7464	. . . . . . 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 20018	. . . . . . 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 2783	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))) = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))))
96	95	oveq2d 7464	. . . . 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 2788	. . . 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 7456	. . . . . 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 2829	. . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐻‘𝑥) = (𝐻‘𝑦))
103		sneq 4658	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
104	103	xpeq1d 5729	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝑁) = ({𝑦} × 𝑁))
105	104	reseq2d 6009	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ ({𝑥} × 𝑁)) = (𝐹 ↾ ({𝑦} × 𝑁)))
106	105	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁))) = (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁))))
107	102, 106	oveq12d 7466	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) = ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))
108	107	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐻‘𝑥) − (𝐺 Σ_g (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿 ↔ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) ∈ 𝐿))
109	108	rspccva 3634	. . . . . . . 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 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))) ∈ 𝐿)
111	110	fmpttd 7149	. . . . . 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 8898	. . . . . 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 3636	. . . 4 ⊢ (𝜑 → (𝐺 Σ_g (𝑦 ∈ 𝐾 ↦ ((𝐻‘𝑦) − (𝐺 Σ_g (𝐹 ↾ ({𝑦} × 𝑁)))))) ∈ 𝑇)
116	97, 115	eqeltrrd 2845	. . 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 7455	. . . . 5 ⊢ (𝑐 = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) → (𝑐 + 𝑑) = ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑))
119	118	eleq1d 2829	. . . 4 ⊢ (𝑐 = (𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) → ((𝑐 + 𝑑) ∈ 𝑈 ↔ ((𝐺 Σ_g (𝐹 ↾ (𝐾 × 𝑁))) + 𝑑) ∈ 𝑈))
120		oveq2 7456	. . . . 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 2829	. . . 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 3647	. . 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 2845	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐻 ↾ 𝐾)) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 {csn 4648 ↦ cmpt 5249 × cxp 5698 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∘_f cof 7712 ↑_m cmap 8884 Fincfn 9003 Basecbs 17258 +_gcplusg 17311 TopOpenctopn 17481 0_gc0g 17499 Σ_g cgsu 17500 Mndcmnd 18772 Grpcgrp 18973 -_gcsg 18975 CMndccmn 19822 Abelcabl 19823 TopGrpctgp 24100 tsums ctsu 24155

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-tgp 24102

This theorem is referenced by: tsmsxp 24184

W3C validator