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Theorem tmdgsum 24119
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)) ∈ ((𝐽ko 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 7439 . . . . . . . 8 (𝑤 = ∅ → (𝐵m 𝑤) = (𝐵m ∅))
21mpteq1d 5243 . . . . . . 7 (𝑤 = ∅ → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)))
3 xpeq1 5703 . . . . . . . . . 10 (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4 0xp 5787 . . . . . . . . . 10 (∅ × {𝐽}) = ∅
53, 4eqtrdi 2791 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
65fveq2d 6911 . . . . . . . 8 (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
76oveq1d 7446 . . . . . . 7 (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
82, 7eleq12d 2833 . . . . . 6 (𝑤 = ∅ → ((𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽)))
98imbi2d 340 . . . . 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 7439 . . . . . . . 8 (𝑤 = 𝑦 → (𝐵m 𝑤) = (𝐵m 𝑦))
1110mpteq1d 5243 . . . . . . 7 (𝑤 = 𝑦 → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)))
12 xpeq1 5703 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
1312fveq2d 6911 . . . . . . . 8 (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
1413oveq1d 7446 . . . . . . 7 (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
1511, 14eleq12d 2833 . . . . . 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 340 . . . . 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 7439 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵m 𝑤) = (𝐵m (𝑦 ∪ {𝑧})))
1817mpteq1d 5243 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19 xpeq1 5703 . . . . . . . . 9 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
2019fveq2d 6911 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
2120oveq1d 7446 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
2218, 21eleq12d 2833 . . . . . 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 340 . . . . 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 7439 . . . . . . . 8 (𝑤 = 𝐴 → (𝐵m 𝑤) = (𝐵m 𝐴))
2524mpteq1d 5243 . . . . . . 7 (𝑤 = 𝐴 → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)))
26 xpeq1 5703 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
2726fveq2d 6911 . . . . . . . 8 (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
2827oveq1d 7446 . . . . . . 7 (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
2925, 28eleq12d 2833 . . . . . 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 340 . . . . 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 8904 . . . . . . . . . 10 (𝑥 ∈ (𝐵m ∅) → 𝑥 Fn ∅)
32 fn0 6700 . . . . . . . . . 10 (𝑥 Fn ∅ ↔ 𝑥 = ∅)
3331, 32sylib 218 . . . . . . . . 9 (𝑥 ∈ (𝐵m ∅) → 𝑥 = ∅)
3433oveq2d 7447 . . . . . . . 8 (𝑥 ∈ (𝐵m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35 eqid 2735 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
3635gsum0 18710 . . . . . . . 8 (𝐺 Σg ∅) = (0g𝐺)
3734, 36eqtrdi 2791 . . . . . . 7 (𝑥 ∈ (𝐵m ∅) → (𝐺 Σg 𝑥) = (0g𝐺))
3837mpteq2ia 5251 . . . . . 6 (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m ∅) ↦ (0g𝐺))
39 0ex 5313 . . . . . . . 8 ∅ ∈ V
40 tmdgsum.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
41 tmdgsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4240, 41tmdtopon 24105 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4342adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
444fveq2i 6910 . . . . . . . . . 10 (∏t‘(∅ × {𝐽})) = (∏t‘∅)
4544eqcomi 2744 . . . . . . . . 9 (∏t‘∅) = (∏t‘(∅ × {𝐽}))
4645pttoponconst 23621 . . . . . . . 8 ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵m ∅)))
4739, 43, 46sylancr 587 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵m ∅)))
48 tmdmnd 24099 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
4948adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
5041, 35mndidcl 18775 . . . . . . . 8 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
5149, 50syl 17 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g𝐺) ∈ 𝐵)
5247, 43, 51cnmptc 23686 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵m ∅) ↦ (0g𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
5338, 52eqeltrid 2843 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54 oveq2 7439 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
5554cbvmptv 5261 . . . . . . . . . 10 (𝑥 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56 eqid 2735 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
57 simpl1l 1223 . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59 snfi 9082 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
60 unfi 9210 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
6158, 59, 60sylancl 586 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
6261adantr 480 . . . . . . . . . . . 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 8888 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐵m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
6463adantl 481 . . . . . . . . . . . 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 fvexd 6922 . . . . . . . . . . . . 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)
6664, 62, 65fdmfifsupp 9413 . . . . . . . . . . . 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))
67 simpl2r 1226 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵m (𝑦 ∪ {𝑧}))) → ¬ 𝑧𝑦)
68 disjsn 4716 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
6967, 68sylibr 234 . . . . . . . . . . . 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}) = ∅)
70 eqidd 2736 . . . . . . . . . . . 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}))
7141, 35, 56, 57, 62, 64, 66, 69, 70gsumsplit 19961 . . . . . . . . . . 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}))))
7271mpteq2dva 5248 . . . . . . . . . 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})))))
7355, 72eqtrid 2787 . . . . . . . . 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})))))
74 simp1r 1197 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐺 ∈ TopMnd)
7574, 42syl 17 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
76 eqid 2735 . . . . . . . . . . . 12 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
7776pttoponconst 23621 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵m (𝑦 ∪ {𝑧}))))
7861, 75, 77syl2anc 584 . . . . . . . . . 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}))))
79 toponuni 22936 . . . . . . . . . . . . . 14 ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵m (𝑦 ∪ {𝑧}))) → (𝐵m (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8078, 79syl 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})))
8180mpteq1d 5243 . . . . . . . . . . . 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)))
82 topontop 22935 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
8374, 42, 823syl 18 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
84 fconst6g 6798 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
8583, 84syl 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)
86 ssun1 4188 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
8786a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
88 eqid 2735 . . . . . . . . . . . . . 14 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89 xpssres 6038 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
9086, 89ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
9190eqcomi 2744 . . . . . . . . . . . . . . 15 (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
9291fveq2i 6910 . . . . . . . . . . . . . 14 (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
9388, 76, 92ptrescn 23663 . . . . . . . . . . . . 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}))))
9461, 85, 87, 93syl3anc 1370 . . . . . . . . . . . 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}))))
9581, 94eqeltrd 2839 . . . . . . . . . . 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}))))
96 eqid 2735 . . . . . . . . . . . . 13 (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
9796pttoponconst 23621 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵m 𝑦)))
9858, 75, 97syl2anc 584 . . . . . . . . . . 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)))
99 simp3 1137 . . . . . . . . . . 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))
100 oveq2 7439 . . . . . . . . . . 11 (𝑥 = (𝑤𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤𝑦)))
10178, 95, 98, 99, 100cnmpt11 23687 . . . . . . . . . 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))
10264feqmptd 6977 . . . . . . . . . . . . . . . 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)))
103102reseq1d 5999 . . . . . . . . . . . . . . 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}))
104 ssun2 4189 . . . . . . . . . . . . . . . 16 {𝑧} ⊆ (𝑦 ∪ {𝑧})
105 resmpt 6057 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
106104, 105ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))
107103, 106eqtrdi 2791 . . . . . . . . . . . . . 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)))
108107oveq2d 7447 . . . . . . . . . . . . 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))))
109 cmnmnd 19830 . . . . . . . . . . . . . . 15 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
11057, 109syl 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)
111 vex 3482 . . . . . . . . . . . . . . 15 𝑧 ∈ V
112111a1i 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)
113 vsnid 4668 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
114 elun2 4193 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
115113, 114mp1i 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}))
11664, 115ffvelcdmd 7105 . . . . . . . . . . . . . 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)
117 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑤𝑘) = (𝑤𝑧))
11841, 117gsumsn 19987 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
119110, 112, 116, 118syl3anc 1370 . . . . . . . . . . . . 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))
120108, 119eqtrd 2775 . . . . . . . . . . . 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))
121120mpteq2dva 5248 . . . . . . . . . . 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)))
12280mpteq1d 5243 . . . . . . . . . . . . 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)))
123113, 114mp1i 13 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
12488, 76ptpjcn 23635 . . . . . . . . . . . . . 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)))
12561, 85, 123, 124syl3anc 1370 . . . . . . . . . . . . 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)))
126122, 125eqeltrd 2839 . . . . . . . . . . . 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)))
127 fvconst2g 7222 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
12883, 123, 127syl2anc 584 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
129128oveq2d 7447 . . . . . . . . . . . 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))
130126, 129eleqtrd 2841 . . . . . . . . . . 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))
131121, 130eqeltrd 2839 . . . . . . . . . 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))
13240, 56, 74, 78, 101, 131cnmpt1plusg 24111 . . . . . . . . 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))
13373, 132eqeltrd 2839 . . . . . . . 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))
1341333expia 1120 . . . . . . 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)))
135134expcom 413 . . . . . 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))))
136135a2d 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))))
1379, 16, 23, 30, 53, 136findcard2s 9204 . . . 4 (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
138137com12 32 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
1391383impia 1116 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
14042, 82syl 17 . . . . 5 (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
141 xkopt 23679 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142140, 141sylan 580 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1431423adant1 1129 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144143oveq1d 7446 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145139, 144eleqtrrd 2842 1 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  tcpt 17485  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  CMndccmn 19813  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  ko cxko 23585  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-rest 17469  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-mre 17631  df-mrc 17632  df-acs 17634  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-xko 23587  df-tmd 24096
This theorem is referenced by:  tmdgsum2  24120
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