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Theorem tsmsxp 22005
Description: Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18421 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22003 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 21930 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
31, 2syl 17 . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
433ad2ant1 1102 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝐺 ∈ TopMnd)
5 simp2 1082 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑢 ∈ (TopOpen‘𝐺))
6 eqid 2651 . . . . . . . . . . . . 13 (TopOpen‘𝐺) = (TopOpen‘𝐺)
7 tsmsxp.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐺)
86, 7tmdtopon 21932 . . . . . . . . . . . 12 (𝐺 ∈ TopMnd → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
94, 8syl 17 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (TopOpen‘𝐺) ∈ (TopOn‘𝐵))
10 toponss 20779 . . . . . . . . . . 11 (((TopOpen‘𝐺) ∈ (TopOn‘𝐵) ∧ 𝑢 ∈ (TopOpen‘𝐺)) → 𝑢𝐵)
119, 5, 10syl2anc 694 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑢𝐵)
12 simp3 1083 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑥𝑢)
1311, 12sseldd 3637 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝑥𝐵)
14 tmdmnd 21926 . . . . . . . . . . 11 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
154, 14syl 17 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → 𝐺 ∈ Mnd)
16 eqid 2651 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
177, 16mndidcl 17355 . . . . . . . . . 10 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
1815, 17syl 17 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (0g𝐺) ∈ 𝐵)
19 eqid 2651 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
207, 19, 16mndrid 17359 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
2115, 13, 20syl2anc 694 . . . . . . . . . 10 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
2221, 12eqeltrd 2730 . . . . . . . . 9 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → (𝑥(+g𝐺)(0g𝐺)) ∈ 𝑢)
237, 6, 19tmdcn2 21940 . . . . . . . . 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 1373 . . . . . . . 8 ((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) → ∃𝑣 ∈ (TopOpen‘𝐺)∃𝑡 ∈ (TopOpen‘𝐺)(𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢))
25 r19.29 3101 . . . . . . . . 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 1117 . . . . . . . . . . . . . . . 16 (((𝜑𝑢 ∈ (TopOpen‘𝐺) ∧ 𝑥𝑢) ∧ (𝑣 ∈ (TopOpen‘𝐺) ∧ 𝑡 ∈ (TopOpen‘𝐺)) ∧ (𝑥𝑣 ∧ (0g𝐺) ∈ 𝑡 ∧ ∀𝑐𝑣𝑑𝑡 (𝑐(+g𝐺)𝑑) ∈ 𝑢)) → 𝑥𝑣)
27 elfpw 8309 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝑦 ⊆ (𝐴 × 𝐶) ∧ 𝑦 ∈ Fin))
2827simplbi 475 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (𝐴 × 𝐶))
2928ad2antrl 764 . . . . . . . . . . . . . . . . . . . . 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 5355 . . . . . . . . . . . . . . . . . . . . 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 5600 . . . . . . . . . . . . . . . . . . . 20 dom (𝐴 × 𝐶) ⊆ 𝐴
3331, 32syl6ss 3648 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ∈ Fin)
3534ad2antrl 764 . . . . . . . . . . . . . . . . . . . 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 8285 . . . . . . . . . . . . . . . . . . . 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 8309 . . . . . . . . . . . . . . . . . . 19 (dom 𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (dom 𝑦𝐴 ∧ dom 𝑦 ∈ Fin))
3933, 37, 38sylanbrc 699 . . . . . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (.g𝐺) = (.g𝐺)
41 simpl11 1156 . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . 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 8309 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑏𝐴𝑏 ∈ Fin))
4746simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . . . . 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 13185 . . . . . . . . . . . . . . . . . . . . . . . . . 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 17614 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Mnd ∧ (#‘𝑏) ∈ ℕ0) → ((#‘𝑏)(.g𝐺)(0g𝐺)) = (0g𝐺))
5550, 53, 54syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . . . . . . . . . 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 21947 . . . . . . . . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . . . . . . . 25 (-g𝐺) = (-g𝐺)
73 simp3l 1109 . . . . . . . . . . . . . . . . . . . . . . . . 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 1154 . . . . . . . . . . . . . . . . . . . . . . . . 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 1150 . . . . . . . . . . . . . . . . . . . . . . . . 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 1151 . . . . . . . . . . . . . . . . . . . . . . . . 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 1148 . . . . . . . . . . . . . . . . . . . . . . . . 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 22003 . . . . . . . . . . . . . . . . . . . . . . . 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 1101 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1152 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1150 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1154 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Rel (𝐴 × 𝐶)
97 relss 5240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ⊆ (𝐴 × 𝐶) → (Rel (𝐴 × 𝐶) → Rel 𝑦))
9828, 96, 97mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → Rel 𝑦)
99 relssdmrn 5694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Rel 𝑦𝑦 ⊆ (dom 𝑦 × ran 𝑦))
10098, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝑦 ⊆ (dom 𝑦 × ran 𝑦))
101 xpss12 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((dom 𝑦𝑏 ∧ ran 𝑦𝑛) → (dom 𝑦 × ran 𝑦) ⊆ (𝑏 × 𝑛))
102100, 101sylan9ss 3649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ∧ (dom 𝑦𝑏 ∧ ran 𝑦𝑛)) → 𝑦 ⊆ (𝑏 × 𝑛))
10392, 93, 95, 102syl12anc 1364 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑛𝐶𝑛 ∈ Fin))
106 xpss12 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑏𝐴𝑛𝐶) → (𝑏 × 𝑛) ⊆ (𝐴 × 𝐶))
107 xpfi 8272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑏 ∈ Fin ∧ 𝑛 ∈ Fin) → (𝑏 × 𝑛) ∈ Fin)
108106, 107anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑏𝐴𝑛𝐶) ∧ (𝑏 ∈ Fin ∧ 𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
109108an4s 886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑏𝐴𝑏 ∈ Fin) ∧ (𝑛𝐶𝑛 ∈ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
11046, 105, 109syl2anb 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑛 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑏 × 𝑛) ⊆ (𝐴 × 𝐶) ∧ (𝑏 × 𝑛) ∈ Fin))
111 elfpw 8309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑏 × 𝑛) → (𝑦𝑧𝑦 ⊆ (𝑏 × 𝑛)))
116 reseq2 5423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑏 × 𝑛) → (𝐹𝑧) = (𝐹 ↾ (𝑏 × 𝑛)))
117116oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = (𝑏 × 𝑛) → (𝐺 Σg (𝐹𝑧)) = (𝐺 Σg (𝐹 ↾ (𝑏 × 𝑛))))
118117eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑔 = → (𝐺 Σg 𝑔) = (𝐺 Σg ))
125124eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑔 = → ((𝐺 Σg 𝑔) ∈ 𝑡 ↔ (𝐺 Σg ) ∈ 𝑡))
126125cbvralv 3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22004 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . . . . . 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 1302 . . . . . . . . . . . . . . . . . . . . . . . 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 3064 . . . . . . . . . . . . . . . . . . . . . . 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 1284 . . . . . . . . . . . . . . . . . . . . . 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 3064 . . . . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . . . . . 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 2995 . . . . . . . . . . . . . . . . . 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 3659 . . . . . . . . . . . . . . . . . . . . 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 3015 . . . . . . . . . . . . . . . . . . 19 (𝑎 = dom 𝑦 → (∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(𝑎𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐴 ∩ Fin)(dom 𝑦𝑏 → (𝐺 Σg (𝐻𝑏)) ∈ 𝑢)))
141140rspcev 3340 . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . 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 3061 . . . . . . . . . . . . . . . 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 1286 . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . 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 3060 . . . . . . . . . . . 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 3060 . . . . . . . . 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 710 . . . . . . 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 1286 . . . . . 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 2998 . . . 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 588 . . 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 2651 . . . 4 (𝒫 (𝐴 × 𝐶) ∩ Fin) = (𝒫 (𝐴 × 𝐶) ∩ Fin)
158 tgptps 21931 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
1591, 158syl 17 . . . 4 (𝜑𝐺 ∈ TopSp)
160 xpexg 7002 . . . . 5 ((𝐴𝑉𝐶𝑊) → (𝐴 × 𝐶) ∈ V)
16162, 64, 160syl2anc 694 . . . 4 (𝜑 → (𝐴 × 𝐶) ∈ V)
1627, 6, 157, 42, 159, 161, 66eltsms 21983 . . 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 2651 . . . 4 (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
1647, 6, 163, 42, 159, 62, 68eltsms 21983 . . 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 3642 1 (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  0cn0 11330  #chash 13157  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  -gcsg 17471  .gcmg 17587  CMndccmn 18239  TopOnctopon 20763  TopSpctps 20784  TopMndctmd 21921  TopGrpctgp 21922   tsums ctsu 21976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-rest 16130  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-mre 16293  df-mrc 16294  df-acs 16296  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-ntr 20872  df-nei 20950  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-xko 21414  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tmd 21923  df-tgp 21924  df-tsms 21977
This theorem is referenced by: (None)
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