Theorem tsmsxplem1 24097

Description: Lemma for tsmsxp 24099. (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 = (0g‘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.ks	⊢ (𝜑 → dom 𝐷 ⊆ 𝐾)
tsmsxp.d	⊢ (𝜑 → 𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))

Assertion

Ref	Expression
tsmsxplem1	⊢ (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))

Distinct variable groups:   0 ,𝑘   𝑗,𝑘,𝑛,𝑥,𝐺   𝐵,𝑘   𝐷,𝑗,𝑘,𝑛,𝑥   𝑗,𝐿,𝑛,𝑥   𝐴,𝑗,𝑘,𝑛   𝑗,𝐾,𝑘,𝑛,𝑥   𝑗,𝐻,𝑘,𝑛,𝑥   − ,𝑗,𝑛,𝑥   𝐶,𝑗,𝑘,𝑛   𝑗,𝐹,𝑘,𝑛,𝑥   𝜑,𝑗,𝑘,𝑛

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥,𝑗,𝑛)   𝐶(𝑥)   + (𝑥,𝑗,𝑘,𝑛)   𝐽(𝑥,𝑗,𝑘,𝑛)   𝐿(𝑘)   − (𝑘)   𝑉(𝑥,𝑗,𝑘,𝑛)   𝑊(𝑥,𝑗,𝑘,𝑛)   0 (𝑥,𝑗,𝑛)

Proof of Theorem tsmsxplem1

Dummy variables 𝑔 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsxp.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))
2	1	elin2d 4157	. . 3 ⊢ (𝜑 → 𝐾 ∈ Fin)
3		elfpw 9254	. . . . . . . 8 ⊢ (𝐾 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐾 ⊆ 𝐴 ∧ 𝐾 ∈ Fin))
4	3	simplbi 497	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾 ⊆ 𝐴)
5	1, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ 𝐴)
6	5	sselda 3933	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾) → 𝑗 ∈ 𝐴)
7		tsmsxp.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
8		tsmsxp.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
9		eqid 2736	. . . . . 6 ⊢ (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
10		tsmsxp.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐺 ∈ CMnd)
12		tsmsxp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TopGrp)
13		tgptps 24024	. . . . . . . 8 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TopSp)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐺 ∈ TopSp)
16		tsmsxp.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
18		tsmsxp.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)
19		fovcdm 7528	. . . . . . . . 9 ⊢ ((𝐹:(𝐴 × 𝐶)⟶𝐵 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
20	18, 19	syl3an1 1163	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
21	20	3expa 1118	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
22	21	fmpttd 7060	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)):𝐶⟶𝐵)
23		tsmsxp.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))
24		df-ima 5637	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) “ 𝐿) = ran ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) ↾ 𝐿)
25	8, 7	tgptopon 24026	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵))
26	12, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
27		tsmsxp.l	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ 𝐽)
28		toponss 22871	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐿 ∈ 𝐽) → 𝐿 ⊆ 𝐵)
29	26, 27, 28	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ⊆ 𝐵)
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐿 ⊆ 𝐵)
31	30	resmptd 5999	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) ↾ 𝐿) = (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
32	31	rneqd 5887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) ↾ 𝐿) = ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
33	24, 32	eqtrid 2783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) “ 𝐿) = ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
34		tsmsxp.h	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
35	34	ffvelcdmda 7029	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ 𝐵)
36		tsmsxp.p	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝐺)
37		eqid 2736	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
38		tsmsxp.m	. . . . . . . . . . . . 13 ⊢ − = (-g‘𝐺)
39	7, 36, 37, 38	grpsubval 18915	. . . . . . . . . . . 12 ⊢ (((𝐻‘𝑗) ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → ((𝐻‘𝑗) − 𝑔) = ((𝐻‘𝑗) + ((invg‘𝐺)‘𝑔)))
40	35, 39	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝐻‘𝑗) − 𝑔) = ((𝐻‘𝑗) + ((invg‘𝐺)‘𝑔)))
41	40	mpteq2dva 5191	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) = (𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) + ((invg‘𝐺)‘𝑔))))
42		tgpgrp 24022	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
43	12, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐺 ∈ Grp)
45	7, 37	grpinvcl 18917	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝐵) → ((invg‘𝐺)‘𝑔) ∈ 𝐵)
46	44, 45	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((invg‘𝐺)‘𝑔) ∈ 𝐵)
47	7, 37	grpinvf 18916	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):𝐵⟶𝐵)
48	44, 47	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (invg‘𝐺):𝐵⟶𝐵)
49	48	feqmptd 6902	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (invg‘𝐺) = (𝑔 ∈ 𝐵 ↦ ((invg‘𝐺)‘𝑔)))
50		eqidd 2737	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) = (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)))
51		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑦 = ((invg‘𝐺)‘𝑔) → ((𝐻‘𝑗) + 𝑦) = ((𝐻‘𝑗) + ((invg‘𝐺)‘𝑔)))
52	46, 49, 50, 51	fmptco 7074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∘ (invg‘𝐺)) = (𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) + ((invg‘𝐺)‘𝑔))))
53	41, 52	eqtr4d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) = ((𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∘ (invg‘𝐺)))
54	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐺 ∈ TopGrp)
55	8, 37	grpinvhmeo 24030	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
56	54, 55	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
57		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) = (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦))
58	57, 7, 36, 8	tgplacthmeo 24047	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ (𝐻‘𝑗) ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
59	54, 35, 58	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
60		hmeoco 23716	. . . . . . . . . 10 ⊢ (((invg‘𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽)) → ((𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
61	56, 59, 60	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ ((𝐻‘𝑗) + 𝑦)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
62	53, 61	eqeltrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) ∈ (𝐽Homeo𝐽))
63	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐿 ∈ 𝐽)
64		hmeoima 23709	. . . . . . . 8 ⊢ (((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) ∈ (𝐽Homeo𝐽) ∧ 𝐿 ∈ 𝐽) → ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) “ 𝐿) ∈ 𝐽)
65	62, 63, 64	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝑔 ∈ 𝐵 ↦ ((𝐻‘𝑗) − 𝑔)) “ 𝐿) ∈ 𝐽)
66	33, 65	eqeltrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) ∈ 𝐽)
67		tsmsxp.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
68	7, 67, 38	grpsubid1 18955	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐻‘𝑗) ∈ 𝐵) → ((𝐻‘𝑗) − 0 ) = (𝐻‘𝑗))
69	44, 35, 68	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐻‘𝑗) − 0 ) = (𝐻‘𝑗))
70		tsmsxp.3	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ 𝐿)
71	70	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 0 ∈ 𝐿)
72		ovex 7391	. . . . . . . 8 ⊢ ((𝐻‘𝑗) − 0 ) ∈ V
73		eqid 2736	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) = (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))
74		oveq2 7366	. . . . . . . . 9 ⊢ (𝑔 = 0 → ((𝐻‘𝑗) − 𝑔) = ((𝐻‘𝑗) − 0 ))
75	73, 74	elrnmpt1s 5908	. . . . . . . 8 ⊢ (( 0 ∈ 𝐿 ∧ ((𝐻‘𝑗) − 0 ) ∈ V) → ((𝐻‘𝑗) − 0 ) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
76	71, 72, 75	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐻‘𝑗) − 0 ) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
77	69, 76	eqeltrrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))
78	7, 8, 9, 11, 15, 17, 22, 23, 66, 77	tsmsi 24078	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
79	6, 78	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
80	79	ralrimiva 3128	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ 𝐾 ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
81		sseq1 3959	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑗) → (𝑦 ⊆ 𝑧 ↔ (𝑓‘𝑗) ⊆ 𝑧))
82	81	imbi1d 341	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑗) → ((𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) ↔ ((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))))
83	82	ralbidv 3159	. . . 4 ⊢ (𝑦 = (𝑓‘𝑗) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) ↔ ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))))
84	83	ac6sfi 9184	. . 3 ⊢ ((𝐾 ∈ Fin ∧ ∀𝑗 ∈ 𝐾 ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))) → ∃𝑓(𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))))
85	2, 80, 84	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)))))
86		frn 6669	. . . . . . . . 9 ⊢ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
87	86	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
88		inss1 4189	. . . . . . . 8 ⊢ (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
89	87, 88	sstrdi 3946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ 𝒫 𝐶)
90		sspwuni 5055	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝒫 𝐶 ↔ ∪ ran 𝑓 ⊆ 𝐶)
91	89, 90	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ∪ ran 𝑓 ⊆ 𝐶)
92		tsmsxp.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
93		elfpw 9254	. . . . . . . . . 10 ⊢ (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝐷 ⊆ (𝐴 × 𝐶) ∧ 𝐷 ∈ Fin))
94	93	simplbi 497	. . . . . . . . 9 ⊢ (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ⊆ (𝐴 × 𝐶))
95		rnss 5888	. . . . . . . . 9 ⊢ (𝐷 ⊆ (𝐴 × 𝐶) → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
96	92, 94, 95	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
97		rnxpss 6130	. . . . . . . 8 ⊢ ran (𝐴 × 𝐶) ⊆ 𝐶
98	96, 97	sstrdi 3946	. . . . . . 7 ⊢ (𝜑 → ran 𝐷 ⊆ 𝐶)
99	98	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷 ⊆ 𝐶)
100	91, 99	unssd 4144	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
101	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐾 ∈ Fin)
102		ffn 6662	. . . . . . . . . 10 ⊢ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → 𝑓 Fn 𝐾)
103	102	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓 Fn 𝐾)
104		dffn4 6752	. . . . . . . . 9 ⊢ (𝑓 Fn 𝐾 ↔ 𝑓:𝐾–onto→ran 𝑓)
105	103, 104	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓:𝐾–onto→ran 𝑓)
106		fofi 9213	. . . . . . . 8 ⊢ ((𝐾 ∈ Fin ∧ 𝑓:𝐾–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
107	101, 105, 106	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
108		inss2 4190	. . . . . . . 8 ⊢ (𝒫 𝐶 ∩ Fin) ⊆ Fin
109	87, 108	sstrdi 3946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ Fin)
110		unifi 9244	. . . . . . 7 ⊢ ((ran 𝑓 ∈ Fin ∧ ran 𝑓 ⊆ Fin) → ∪ ran 𝑓 ∈ Fin)
111	107, 109, 110	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ∪ ran 𝑓 ∈ Fin)
112		elinel2 4154	. . . . . . . 8 ⊢ (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ∈ Fin)
113		rnfi 9240	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ran 𝐷 ∈ Fin)
114	92, 112, 113	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐷 ∈ Fin)
115	114	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷 ∈ Fin)
116		unfi 9095	. . . . . 6 ⊢ ((∪ ran 𝑓 ∈ Fin ∧ ran 𝐷 ∈ Fin) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ Fin)
117	111, 115, 116	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ Fin)
118		elfpw 9254	. . . . 5 ⊢ ((∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((∪ ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶 ∧ (∪ ran 𝑓 ∪ ran 𝐷) ∈ Fin))
119	100, 117, 118	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
120	119	adantrr 717	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
121		ssun2 4131	. . . 4 ⊢ ran 𝐷 ⊆ (∪ ran 𝑓 ∪ ran 𝐷)
122	121	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))) → ran 𝐷 ⊆ (∪ ran 𝑓 ∪ ran 𝐷))
123	119	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
124		fvssunirn 6865	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑗) ⊆ ∪ ran 𝑓
125		ssun1 4130	. . . . . . . . . . . . . 14 ⊢ ∪ ran 𝑓 ⊆ (∪ ran 𝑓 ∪ ran 𝐷)
126	124, 125	sstri 3943	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑗) ⊆ (∪ ran 𝑓 ∪ ran 𝐷)
127		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → 𝑧 = (∪ ran 𝑓 ∪ ran 𝐷))
128	126, 127	sseqtrrid 3977	. . . . . . . . . . . 12 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → (𝑓‘𝑗) ⊆ 𝑧)
129		pm5.5 361	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑗) ⊆ 𝑧 → (((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) ↔ (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
130	128, 129	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → (((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) ↔ (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
131		reseq2 5933	. . . . . . . . . . . . 13 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧) = ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷)))
132	131	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) = (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))))
133	132	eleq1d 2821	. . . . . . . . . . 11 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → ((𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) ↔ (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
134	130, 133	bitrd 279	. . . . . . . . . 10 ⊢ (𝑧 = (∪ ran 𝑓 ∪ ran 𝐷) → (((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) ↔ (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
135	134	rspcv 3572	. . . . . . . . 9 ⊢ ((∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
136	123, 135	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
137	10	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
138		cmnmnd 19726	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
139	137, 138	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ Mnd)
140		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗 ∈ 𝐾)
141	117	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ∈ Fin)
142	100	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∪ ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
143	142	sselda 3933	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → 𝑘 ∈ 𝐶)
144	18	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
145	144, 6	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾) → (𝐹:(𝐴 × 𝐶)⟶𝐵 ∧ 𝑗 ∈ 𝐴))
146	19	3expa 1118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:(𝐴 × 𝐶)⟶𝐵 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
147	145, 146	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
148	147	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ 𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
149	143, 148	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ 𝐵)
150	149	fmpttd 7060	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)):(∪ ran 𝑓 ∪ ran 𝐷)⟶𝐵)
151		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))
152		ovexd 7393	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ V)
153	67	fvexi 6848	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
154	153	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 0 ∈ V)
155	151, 141, 152, 154	fsuppmptdm 9279	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) finSupp 0 )
156	7, 67, 137, 141, 150, 155	gsumcl 19844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
157		velsn 4596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑗} ↔ 𝑦 = 𝑗)
158		ovres 7524	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ {𝑗} ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
159	157, 158	sylanbr 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = 𝑗 ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
160		oveq1 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑗 → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
161	160	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = 𝑗 ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
162	159, 161	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑗 ∧ 𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑗𝐹𝑘))
163	162	mpteq2dva 5191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑗 → (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘)) = (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
164	163	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑗 → (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘))) = (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
165	7, 164	gsumsn 19883	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ 𝑗 ∈ 𝐾 ∧ (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) → (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘))))) = (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
166	139, 140, 156, 165	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘))))) = (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
167		snfi 8980	. . . . . . . . . . . . 13 ⊢ {𝑗} ∈ Fin
168	167	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ∈ Fin)
169	18	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
170	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗 ∈ 𝐴)
171	170	snssd 4765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ⊆ 𝐴)
172		xpss12 5639	. . . . . . . . . . . . . 14 ⊢ (({𝑗} ⊆ 𝐴 ∧ (∪ ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶) → ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
173	171, 142, 172	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
174	169, 173	fssresd 6701	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))):({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))⟶𝐵)
175		xpfi 9220	. . . . . . . . . . . . . 14 ⊢ (({𝑗} ∈ Fin ∧ (∪ ran 𝑓 ∪ ran 𝐷) ∈ Fin) → ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
176	167, 141, 175	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
177	174, 176, 154	fdmfifsupp 9278	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))) finSupp 0 )
178	7, 67, 137, 168, 141, 174, 177	gsumxp 19905	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))𝑘))))))
179	142	resmptd 5999	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷)) = (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
180	179	oveq2d 7374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝑘 ∈ (∪ ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
181	166, 178, 180	3eqtr4rd 2782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))))
182	181	eleq1d 2821	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) ↔ (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))
183		ovex 7391	. . . . . . . . . . 11 ⊢ ((𝐻‘𝑗) − 𝑔) ∈ V
184	73, 183	elrnmpti 5911	. . . . . . . . . 10 ⊢ ((𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) ↔ ∃𝑔 ∈ 𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = ((𝐻‘𝑗) − 𝑔))
185		isabl 19713	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
186	43, 10, 185	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Abel)
187	186	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → 𝐺 ∈ Abel)
188	6, 35	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾) → (𝐻‘𝑗) ∈ 𝐵)
189	188	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → (𝐻‘𝑗) ∈ 𝐵)
190	29	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐿 ⊆ 𝐵)
191	190	sselda 3933	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → 𝑔 ∈ 𝐵)
192	7, 38, 187, 189, 191	ablnncan 19749	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → ((𝐻‘𝑗) − ((𝐻‘𝑗) − 𝑔)) = 𝑔)
193		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → 𝑔 ∈ 𝐿)
194	192, 193	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → ((𝐻‘𝑗) − ((𝐻‘𝑗) − 𝑔)) ∈ 𝐿)
195		oveq2 7366	. . . . . . . . . . . . 13 ⊢ ((𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = ((𝐻‘𝑗) − 𝑔) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) = ((𝐻‘𝑗) − ((𝐻‘𝑗) − 𝑔)))
196	195	eleq1d 2821	. . . . . . . . . . . 12 ⊢ ((𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = ((𝐻‘𝑗) − 𝑔) → (((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻‘𝑗) − ((𝐻‘𝑗) − 𝑔)) ∈ 𝐿))
197	194, 196	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔 ∈ 𝐿) → ((𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = ((𝐻‘𝑗) − 𝑔) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
198	197	rexlimdva 3137	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∃𝑔 ∈ 𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = ((𝐻‘𝑗) − 𝑔) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
199	184, 198	biimtrid 242	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
200	182, 199	sylbid 240	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ (∪ ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔)) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
201	136, 200	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
202	201	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑗 ∈ 𝐾) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
203	202	ralimdva 3148	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))) → ∀𝑗 ∈ 𝐾 ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
204	203	impr 454	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))) → ∀𝑗 ∈ 𝐾 ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
205		fveq2 6834	. . . . . . 7 ⊢ (𝑗 = 𝑥 → (𝐻‘𝑗) = (𝐻‘𝑥))
206		sneq 4590	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → {𝑗} = {𝑥})
207	206	xpeq1d 5653	. . . . . . . . 9 ⊢ (𝑗 = 𝑥 → ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)) = ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷)))
208	207	reseq2d 5938	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))) = (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))
209	208	oveq2d 7374	. . . . . . 7 ⊢ (𝑗 = 𝑥 → (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷)))))
210	205, 209	oveq12d 7376	. . . . . 6 ⊢ (𝑗 = 𝑥 → ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) = ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))))
211	210	eleq1d 2821	. . . . 5 ⊢ (𝑗 = 𝑥 → (((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
212	211	cbvralvw 3214	. . . 4 ⊢ (∀𝑗 ∈ 𝐾 ((𝐻‘𝑗) − (𝐺 Σg (𝐹 ↾ ({𝑗} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
213	204, 212	sylib 218	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))) → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
214		sseq2 3960	. . . . 5 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → (ran 𝐷 ⊆ 𝑛 ↔ ran 𝐷 ⊆ (∪ ran 𝑓 ∪ ran 𝐷)))
215		xpeq2 5645	. . . . . . . . . 10 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → ({𝑥} × 𝑛) = ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷)))
216	215	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → (𝐹 ↾ ({𝑥} × 𝑛)) = (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))
217	216	oveq2d 7374	. . . . . . . 8 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷)))))
218	217	oveq2d 7374	. . . . . . 7 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) = ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))))
219	218	eleq1d 2821	. . . . . 6 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → (((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
220	219	ralbidv 3159	. . . . 5 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → (∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
221	214, 220	anbi12d 632	. . . 4 ⊢ (𝑛 = (∪ ran 𝑓 ∪ ran 𝐷) → ((ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿) ↔ (ran 𝐷 ⊆ (∪ ran 𝑓 ∪ ran 𝐷) ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)))
222	221	rspcev 3576	. . 3 ⊢ (((∪ ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝐷 ⊆ (∪ ran 𝑓 ∪ ran 𝐷) ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × (∪ ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
223	120, 122, 213, 222	syl12anc 836	. 2 ⊢ ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗 ∈ 𝐾 ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓‘𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔 ∈ 𝐿 ↦ ((𝐻‘𝑗) − 𝑔))))) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
224	85, 223	exlimddv 1936	1 ⊢ (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  Basecbs 17136  +_gcplusg 17177  TopOpenctopn 17341  0_gc0g 17359   Σ_g cgsu 17360  Mndcmnd 18659  Grpcgrp 18863  inv_gcminusg 18864  -_gcsg 18865  CMndccmn 19709  Abelcabl 19710  TopOnctopon 22854  TopSpctps 22876  Homeochmeo 23697  TopGrpctgp 24015   tsums ctsu 24070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-gsum 17362  df-topgen 17363  df-mre 17505  df-mrc 17506  df-acs 17508  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-abl 19712  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-ntr 22964  df-nei 23042  df-cn 23171  df-cnp 23172  df-tx 23506  df-hmeo 23699  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-tmd 24016  df-tgp 24017  df-tsms 24071

This theorem is referenced by:  tsmsxp  24099