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Theorem tmdgsum 21822
Description: In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
Hypotheses
Ref Expression
tmdgsum.j 𝐽 = (TopOpen‘𝐺)
tmdgsum.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
tmdgsum ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐵   𝑥,𝐺

Proof of Theorem tmdgsum
Dummy variables 𝑘 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6618 . . . . . . . 8 (𝑤 = ∅ → (𝐵𝑚 𝑤) = (𝐵𝑚 ∅))
21mpteq1d 4703 . . . . . . 7 (𝑤 = ∅ → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)))
3 xpeq1 5093 . . . . . . . . . 10 (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4 0xp 5165 . . . . . . . . . 10 (∅ × {𝐽}) = ∅
53, 4syl6eq 2671 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
65fveq2d 6157 . . . . . . . 8 (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
76oveq1d 6625 . . . . . . 7 (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
82, 7eleq12d 2692 . . . . . 6 (𝑤 = ∅ → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽)))
98imbi2d 330 . . . . 5 (𝑤 = ∅ → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))))
10 oveq2 6618 . . . . . . . 8 (𝑤 = 𝑦 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝑦))
1110mpteq1d 4703 . . . . . . 7 (𝑤 = 𝑦 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)))
12 xpeq1 5093 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
1312fveq2d 6157 . . . . . . . 8 (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
1413oveq1d 6625 . . . . . . 7 (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
1511, 14eleq12d 2692 . . . . . 6 (𝑤 = 𝑦 → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)))
1615imbi2d 330 . . . . 5 (𝑤 = 𝑦 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))))
17 oveq2 6618 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵𝑚 𝑤) = (𝐵𝑚 (𝑦 ∪ {𝑧})))
1817mpteq1d 4703 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19 xpeq1 5093 . . . . . . . . 9 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
2019fveq2d 6157 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
2120oveq1d 6625 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
2218, 21eleq12d 2692 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
2322imbi2d 330 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
24 oveq2 6618 . . . . . . . 8 (𝑤 = 𝐴 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝐴))
2524mpteq1d 4703 . . . . . . 7 (𝑤 = 𝐴 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)))
26 xpeq1 5093 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
2726fveq2d 6157 . . . . . . . 8 (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
2827oveq1d 6625 . . . . . . 7 (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
2925, 28eleq12d 2692 . . . . . 6 (𝑤 = 𝐴 → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
3029imbi2d 330 . . . . 5 (𝑤 = 𝐴 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))))
31 elmapfn 7832 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 Fn ∅)
32 fn0 5973 . . . . . . . . . 10 (𝑥 Fn ∅ ↔ 𝑥 = ∅)
3331, 32sylib 208 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 = ∅)
3433oveq2d 6626 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35 eqid 2621 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
3635gsum0 17210 . . . . . . . 8 (𝐺 Σg ∅) = (0g𝐺)
3734, 36syl6eq 2671 . . . . . . 7 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (0g𝐺))
3837mpteq2ia 4705 . . . . . 6 (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺))
39 0ex 4755 . . . . . . . 8 ∅ ∈ V
40 tmdgsum.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
41 tmdgsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4240, 41tmdtopon 21808 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4342adantl 482 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
444fveq2i 6156 . . . . . . . . . 10 (∏t‘(∅ × {𝐽})) = (∏t‘∅)
4544eqcomi 2630 . . . . . . . . 9 (∏t‘∅) = (∏t‘(∅ × {𝐽}))
4645pttoponconst 21323 . . . . . . . 8 ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
4739, 43, 46sylancr 694 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
48 tmdmnd 21802 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
4948adantl 482 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
5041, 35mndidcl 17240 . . . . . . . 8 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
5149, 50syl 17 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g𝐺) ∈ 𝐵)
5247, 43, 51cnmptc 21388 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
5338, 52syl5eqel 2702 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54 oveq2 6618 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
5554cbvmptv 4715 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56 eqid 2621 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
57 simpl1l 1110 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝐺 ∈ CMnd)
58 simp2l 1085 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59 snfi 7990 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
60 unfi 8179 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
6158, 59, 60sylancl 693 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
6261adantr 481 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) ∈ Fin)
63 elmapi 7831 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
6463adantl 482 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
65 fvex 6163 . . . . . . . . . . . . . 14 (0g𝐺) ∈ V
6665a1i 11 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (0g𝐺) ∈ V)
6764, 62, 66fdmfifsupp 8237 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤 finSupp (0g𝐺))
68 simpl2r 1113 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → ¬ 𝑧𝑦)
69 disjsn 4221 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
7068, 69sylibr 224 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∩ {𝑧}) = ∅)
71 eqidd 2622 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
7241, 35, 56, 57, 62, 64, 67, 70, 71gsumsplit 18260 . . . . . . . . . . 11 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg 𝑤) = ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧}))))
7372mpteq2dva 4709 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
7455, 73syl5eq 2667 . . . . . . . . 9 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
75 simp1r 1084 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐺 ∈ TopMnd)
7675, 42syl 17 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
77 eqid 2621 . . . . . . . . . . . 12 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
7877pttoponconst 21323 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))))
7961, 76, 78syl2anc 692 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))))
80 toponuni 20651 . . . . . . . . . . . . . 14 ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐵𝑚 (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8179, 80syl 17 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝐵𝑚 (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8281mpteq1d 4703 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑦)) = (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)))
83 topontop 20650 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
8475, 42, 833syl 18 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
85 fconst6g 6056 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
8684, 85syl 17 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
87 ssun1 3759 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
8887a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
89 eqid 2621 . . . . . . . . . . . . . 14 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
90 xpssres 5398 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
9187, 90ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
9291eqcomi 2630 . . . . . . . . . . . . . . 15 (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
9392fveq2i 6156 . . . . . . . . . . . . . 14 (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
9489, 77, 93ptrescn 21365 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
9561, 86, 88, 94syl3anc 1323 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
9682, 95eqeltrd 2698 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
97 eqid 2621 . . . . . . . . . . . . 13 (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
9897pttoponconst 21323 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵𝑚 𝑦)))
9958, 76, 98syl2anc 692 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵𝑚 𝑦)))
100 simp3 1061 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
101 oveq2 6618 . . . . . . . . . . 11 (𝑥 = (𝑤𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤𝑦)))
10279, 96, 99, 100, 101cnmpt11 21389 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤𝑦))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
10364feqmptd 6211 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤 = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)))
104103reseq1d 5360 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}))
105 ssun2 3760 . . . . . . . . . . . . . . . 16 {𝑧} ⊆ (𝑦 ∪ {𝑧})
106 resmpt 5413 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
107105, 106ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))
108104, 107syl6eq 2671 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
109108oveq2d 6626 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))))
110 cmnmnd 18140 . . . . . . . . . . . . . . 15 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
11157, 110syl 17 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝐺 ∈ Mnd)
112 vex 3192 . . . . . . . . . . . . . . 15 𝑧 ∈ V
113112a1i 11 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑧 ∈ V)
114 vsnid 4185 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
115 elun2 3764 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
116114, 115mp1i 13 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
11764, 116ffvelrnd 6321 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤𝑧) ∈ 𝐵)
118 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑤𝑘) = (𝑤𝑧))
11941, 118gsumsn 18286 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
120111, 113, 117, 119syl3anc 1323 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
121109, 120eqtrd 2655 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝑤𝑧))
122121mpteq2dva 4709 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)))
12381mpteq1d 4703 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) = (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)))
124114, 115mp1i 13 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
12589, 77ptpjcn 21337 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
12661, 86, 124, 125syl3anc 1323 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
127123, 126eqeltrd 2698 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
128 fvconst2g 6427 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
12984, 124, 128syl2anc 692 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
130129oveq2d 6626 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
131127, 130eleqtrd 2700 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
132122, 131eqeltrd 2698 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
13340, 56, 75, 79, 102, 132cnmpt1plusg 21814 . . . . . . . . 9 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
13474, 133eqeltrd 2698 . . . . . . . 8 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
1351343expia 1264 . . . . . . 7 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
136135expcom 451 . . . . . 6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → ((𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
137136a2d 29 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
1389, 16, 23, 30, 53, 137findcard2s 8153 . . . 4 (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
139138com12 32 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
1401393impia 1258 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
14142, 83syl 17 . . . . 5 (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
142 xkopt 21381 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143141, 142sylan 488 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1441433adant1 1077 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
145144oveq1d 6625 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
146140, 145eleqtrrd 2701 1 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678   × cxp 5077  cres 5081   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Fincfn 7907  Basecbs 15792  +gcplusg 15873  TopOpenctopn 16014  tcpt 16031  0gc0g 16032   Σg cgsu 16033  Mndcmnd 17226  CMndccmn 18125  Topctop 20630  TopOnctopon 20647   Cn ccn 20951   ^ko cxko 21287  TopMndctmd 21797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-fi 8269  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-rest 16015  df-0g 16034  df-gsum 16035  df-topgen 16036  df-pt 16037  df-mre 16178  df-mrc 16179  df-acs 16181  df-plusf 17173  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-mulg 17473  df-cntz 17682  df-cmn 18127  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cn 20954  df-cnp 20955  df-cmp 21113  df-tx 21288  df-xko 21289  df-tmd 21799
This theorem is referenced by:  tmdgsum2  21823
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