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Theorem tsmsxp 22159
Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18575 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22157 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
Hypotheses
Ref Expression
tsmsxp.b 𝐵 = (Base‘𝐺)
tsmsxp.g (𝜑𝐺 ∈ CMnd)
tsmsxp.2 (𝜑𝐺 ∈ TopGrp)
tsmsxp.a (𝜑𝐴𝑉)
tsmsxp.c (𝜑𝐶𝑊)
tsmsxp.f (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
tsmsxp.h (𝜑𝐻:𝐴𝐵)
tsmsxp.1 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
Assertion
Ref Expression
tsmsxp (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))
Distinct variable groups:   𝑗,𝑘,𝐺   𝐵,𝑘   𝐴,𝑗,𝑘   𝑗,𝐻,𝑘   𝐶,𝑗,𝑘   𝑗,𝐹,𝑘   𝜑,𝑗,𝑘
Allowed substitution hints:   𝐵(𝑗)   𝑉(𝑗,𝑘)   𝑊(𝑗,𝑘)

Proof of Theorem tsmsxp
Dummy variables 𝑔 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 𝑛 𝑠 𝑡 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsxp.2 . . . . . . . . . . 11 (𝜑𝐺 ∈ TopGrp)
2 tgptmd 22084 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
31, 2syl 17 . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
433ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝐺 ∈ TopMnd)
5 simp2 1132 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6 eqid 2760 . . . . . . . . . . . . 13 (TopOpen‘𝐺) = (TopOpen‘𝐺)
7 tsmsxp.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐺)
86, 7tmdtopon 22086 . . . . . . . . . . . 12 (𝐺 ∈ TopMnd → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
94, 8syl 17 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
10 toponss 20933 . . . . . . . . . . 11 (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢𝐵)
119, 5, 10syl2anc 696 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑢𝐵)
12 simp3 1133 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑥𝑢)
1311, 12sseldd 3745 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑥𝐵)
14 tmdmnd 22080 . . . . . . . . . . 11 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
154, 14syl 17 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝐺 ∈ Mnd)
16 eqid 2760 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
177, 16mndidcl 17509 . . . . . . . . . 10 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
1815, 17syl 17 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (0g𝐺) ∈ 𝐵)
19 eqid 2760 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
207, 19, 16mndrid 17513 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
2115, 13, 20syl2anc 696 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
2221, 12eqeltrd 2839 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (𝑥(+g𝐺)(0g𝐺)) ∈ 𝑢)
237, 6, 19tmdcn2 22094 . . . . . . . . 9 (((𝐺 ∈ TopMnd ∧ 𝑢 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝐵 ∧ (0g𝐺) ∈ 𝐵 ∧ (𝑥(+g𝐺)(0g𝐺)) ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢))
244, 5, 13, 18, 22, 23syl23anc 1484 . . . . . . . 8 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢))
25 r19.29 3210 . . . . . . . . 9 ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘𝐺)((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)))
26 simp31 1252 . . . . . . . . . . . . . . . 16 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → 𝑥𝑣)
27 elfpw 8433 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
2827simplbi 478 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
2928ad2antrl 766 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → 𝑦 ⊆ (𝐴 × 𝐶))
30 dmss 5478 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ⊆ (𝐴 × 𝐶) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
3129, 30syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → dom 𝑦 ⊆ dom (𝐴 × 𝐶))
32 dmxpss 5723 . . . . . . . . . . . . . . . . . . . 20 dom (𝐴 × 𝐶) ⊆ 𝐴
3331, 32syl6ss 3756 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → dom 𝑦𝐴)
3427simprbi 483 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
3534ad2antrl 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → 𝑦 ∈ Fin)
36 dmfi 8409 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ Fin → dom 𝑦 ∈ Fin)
3735, 36syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → dom 𝑦 ∈ Fin)
38 elfpw 8433 . . . . . . . . . . . . . . . . . . 19 (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦𝐴 ∧ dom 𝑦 ∈ Fin))
3933, 37, 38sylanbrc 701 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
40 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . 23 (.g𝐺) = (.g𝐺)
41 simpl11 1315 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝜑)
42 tsmsxp.g . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐺 ∈ CMnd)
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝐺 ∈ CMnd)
4441, 3syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝐺 ∈ TopMnd)
45 simprrl 823 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
46 elfpw 8433 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏𝐴𝑏 ∈ Fin))
4746simprbi 483 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ∈ Fin)
4845, 47syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝑏 ∈ Fin)
49 simpl2r 1285 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝑡 ∈ (TopOpen‘𝐺))
5044, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → 𝐺 ∈ Mnd)
5150, 17syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → (0g𝐺) ∈ 𝐵)
52 hashcl 13339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 ∈ Fin → (♯‘𝑏) ∈ ℕ0)
5348, 52syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → (♯‘𝑏) ∈ ℕ0)
547, 40, 16mulgnn0z 17768 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Mnd ∧ (♯‘𝑏) ∈ ℕ0) → ((♯‘𝑏)(.g𝐺)(0g𝐺)) = (0g𝐺))
5550, 53, 54syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → ((♯‘𝑏)(.g𝐺)(0g𝐺)) = (0g𝐺))
56 simpl32 1329 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → (0g𝐺) ∈ 𝑡)
5755, 56eqeltrd 2839 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → ((♯‘𝑏)(.g𝐺)(0g𝐺)) ∈ 𝑡)
586, 7, 40, 43, 44, 48, 49, 51, 57tmdgsum2 22101 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → ∃𝑠 ∈ (TopOpen‘𝐺)((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))
59 simp111 1387 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝜑)
6059, 42syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐺 ∈ CMnd)
6159, 1syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐺 ∈ TopGrp)
62 tsmsxp.a . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐴𝑉)
6359, 62syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐴𝑉)
64 tsmsxp.c . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐶𝑊)
6559, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐶𝑊)
66 tsmsxp.f . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
6759, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
68 tsmsxp.h . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐻:𝐴𝐵)
6959, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝐻:𝐴𝐵)
70 tsmsxp.1 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
7159, 70sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) ∧ 𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
72 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . . 25 (-g𝐺) = (-g𝐺)
73 simp3l 1244 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝑠 ∈ (TopOpen‘𝐺))
74 simp3rl 1313 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → (0g𝐺) ∈ 𝑠)
75 simp2rl 1309 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
76 simp2rr 1310 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → dom 𝑦𝑏)
77 simp2ll 1307 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
787, 60, 61, 63, 65, 67, 69, 71, 6, 16, 19, 72, 73, 74, 75, 76, 77tsmsxplem1 22157 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
79433adant3 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ CMnd)
80613adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐺 ∈ TopGrp)
81633adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐴𝑉)
82653adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐶𝑊)
83673adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
84693adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝐻:𝐴𝐵)
85413adant3 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝜑)
8685, 70sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) ∧ 𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
87 simp3ll 1311 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑠 ∈ (TopOpen‘𝐺))
88743adant3r 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (0g𝐺) ∈ 𝑠)
89 simp2rl 1309 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
90 simp133 1395 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)
91 simp3rl 1313 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑛 ∈ (𝒫 𝐶 ∩ Fin))
92 simp2ll 1307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
93 simp2rr 1310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → dom 𝑦𝑏)
94 simp3rr 1314 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))
9594simpld 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ran 𝑦𝑛)
96 relxp 5283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Rel (𝐴 × 𝐶)
97 relss 5363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ⊆ (𝐴 × 𝐶) → (Rel (𝐴 × 𝐶) → Rel 𝑦))
9828, 96, 97mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → Rel 𝑦)
99 relssdmrn 5817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Rel 𝑦𝑦 ⊆ (dom 𝑦 × ran 𝑦))
10098, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
101 xpss12 5281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((dom 𝑦𝑏 ∧ ran 𝑦𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
102100, 101sylan9ss 3757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦𝑏 ∧ ran 𝑦𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
10392, 93, 95, 102syl12anc 1475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → 𝑦 ⊆ (𝑏 × 𝑛))
10494simprd 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)
105 elfpw 8433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛𝐶𝑛 ∈ Fin))
106 xpss12 5281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑏𝐴𝑛𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
107 xpfi 8396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
108106, 107anim12i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑏𝐴𝑛𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
109108an4s 904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑏𝐴𝑏 ∈ Fin) ∧ (𝑛𝐶𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
11046, 105, 109syl2anb 497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
111 elfpw 8433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
112110, 111sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
11389, 91, 112syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
114 simp2lr 1308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))
115 sseq2 3768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑏 × 𝑛) → (𝑦𝑧𝑦 ⊆ (𝑏 × 𝑛)))
116 reseq2 5546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑏 × 𝑛) → (𝐹𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
117116oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = (𝑏 × 𝑛) → (𝐺 Σg (𝐹𝑧)) = (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))))
118117eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑏 × 𝑛) → ((𝐺 Σg (𝐹𝑧)) ∈ 𝑣 ↔ (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣))
119115, 118imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑏 × 𝑛) → ((𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣) ↔ (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
120119rspcv 3445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 × 𝑛) ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → (∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣) → (𝑦 ⊆ (𝑏 × 𝑛) → (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)))
121113, 114, 103, 120syl3c 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))) ∈ 𝑣)
122 simp3lr 1312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))
123122simprd 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)
124 oveq2 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑔 = → (𝐺 Σg 𝑔) = (𝐺 Σg ))
125124eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑔 = → ((𝐺 Σg 𝑔) ∈ 𝑡 ↔ (𝐺 Σg ) ∈ 𝑡))
126125cbvralv 3310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡 ↔ ∀ ∈ (𝑠𝑚 𝑏)(𝐺 Σg ) ∈ 𝑡)
127123, 126sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → ∀ ∈ (𝑠𝑚 𝑏)(𝐺 Σg ) ∈ 𝑡)
1287, 79, 80, 81, 82, 83, 84, 86, 6, 16, 19, 72, 87, 88, 89, 90, 91, 103, 104, 121, 127tsmsxplem2 22158 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
1291283exp 1113 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) → (((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
130129exp4a 634 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → (((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) → ((𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡)) → ((𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠)) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
1311303imp1 1441 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) ∧ (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝑦𝑛 ∧ ∀𝑥𝑏 ((𝐻𝑥)(-g𝐺)(𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝑠))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
13278, 131rexlimddv 3173 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
1331323expa 1112 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) ∧ (𝑠 ∈ (TopOpen‘𝐺) ∧ ((0g𝐺) ∈ 𝑠 ∧ ∀𝑔 ∈ (𝑠𝑚 𝑏)(𝐺 Σg 𝑔) ∈ 𝑡))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
13458, 133rexlimddv 3173 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏))) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
135134anassrs 683 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ dom 𝑦𝑏)) → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)
136135expr 644 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))
137136ralrimiva 3104 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))
138 sseq1 3767 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = dom 𝑦 → (𝑎𝑏 ↔ dom 𝑦𝑏))
139138imbi1d 330 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = dom 𝑦 → ((𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢) ↔ (dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
140139ralbidv 3124 . . . . . . . . . . . . . . . . . . 19 (𝑎 = dom 𝑦 → (∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
141140rspcev 3449 . . . . . . . . . . . . . . . . . 18 ((dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))
14239, 137, 141syl2anc 696 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) ∧ (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))
143142rexlimdvaa 3170 . . . . . . . . . . . . . . . 16 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → (∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
14426, 143embantd 59 . . . . . . . . . . . . . . 15 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → ((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
1451443expia 1115 . . . . . . . . . . . . . 14 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺))) → ((𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢) → ((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
146145anassrs 683 . . . . . . . . . . . . 13 ((((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) ∧ 𝑡 ∈ (TopOpen‘𝐺)) → ((𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢) → ((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
147146rexlimdva 3169 . . . . . . . . . . . 12 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢) → ((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
148147com23 86 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → ((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → (∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
149148impd 446 . . . . . . . . . 10 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ 𝑣 ∈ (TopOpen‘𝐺)) → (((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
150149rexlimdva 3169 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (∃𝑣 ∈ (TopOpen‘𝐺)((𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ ∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
15125, 150syl5 34 . . . . . . . 8 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → ((∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) ∧ ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
15224, 151mpan2d 712 . . . . . . 7 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
1531523expia 1115 . . . . . 6 ((𝜑𝑢 ∈ (TopOpen‘𝐺)) → (𝑥𝑢 → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
154153com23 86 . . . . 5 ((𝜑𝑢 ∈ (TopOpen‘𝐺)) → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → (𝑥𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
155154ralrimdva 3107 . . . 4 (𝜑 → (∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)) → ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢))))
156155anim2d 590 . . 3 (𝜑 → ((𝑥𝐵 ∧ ∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣))) → (𝑥𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))))
157 eqid 2760 . . . 4 (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
158 tgptps 22085 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
1591, 158syl 17 . . . 4 (𝜑𝐺 ∈ TopSp)
160 xpexg 7125 . . . . 5 ((𝐴𝑉𝐶𝑊) → (𝐴 × 𝐶) ∈ V)
16162, 64, 160syl2anc 696 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ V)
1627, 6, 157, 42, 159, 161, 66eltsms 22137 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥𝐵 ∧ ∀𝑣 ∈ (TopOpen‘𝐺)(𝑥𝑣 → ∃𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)∀𝑧 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)(𝑦𝑧 → (𝐺 Σg (𝐹𝑧)) ∈ 𝑣)))))
163 eqid 2760 . . . 4 (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
1647, 6, 163, 42, 159, 62, 68eltsms 22137 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐻) ↔ (𝑥𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥𝑢 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))))
165156, 162, 1643imtr4d 283 . 2 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ (𝐺 tsums 𝐻)))
166165ssrdv 3750 1 (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  𝒫 cpw 4302  {csn 4321  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  cres 5268  Rel wrel 5271  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Fincfn 8121  0cn0 11484  chash 13311  Basecbs 16059  +gcplusg 16143  TopOpenctopn 16284  0gc0g 16302   Σg cgsu 16303  Mndcmnd 17495  -gcsg 17625  .gcmg 17741  CMndccmn 18393  TopOnctopon 20917  TopSpctps 20938  TopMndctmd 22075  TopGrpctgp 22076   tsums ctsu 22130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-rest 16285  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-mre 16448  df-mrc 16449  df-acs 16451  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-fbas 19945  df-fg 19946  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-ntr 21026  df-nei 21104  df-cn 21233  df-cnp 21234  df-cmp 21392  df-tx 21567  df-xko 21568  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-tmd 22077  df-tgp 22078  df-tsms 22131
This theorem is referenced by: (None)
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