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Theorem ptcmpfi 23164
Description: A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
Assertion
Ref Expression
ptcmpfi ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)

Proof of Theorem ptcmpfi
Dummy variables 𝑣 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6668 . . . . 5 (𝐹:𝐴⟶Comp → 𝐹 Fn 𝐴)
2 fnresdm 6620 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
31, 2syl 17 . . . 4 (𝐹:𝐴⟶Comp → (𝐹𝐴) = 𝐹)
43adantl 482 . . 3 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (𝐹𝐴) = 𝐹)
54fveq2d 6846 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) = (∏t𝐹))
6 ssid 3966 . . . 4 𝐴𝐴
7 sseq1 3969 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
8 reseq2 5932 . . . . . . . . . 10 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
9 res0 5941 . . . . . . . . . 10 (𝐹 ↾ ∅) = ∅
108, 9eqtrdi 2792 . . . . . . . . 9 (𝑥 = ∅ → (𝐹𝑥) = ∅)
1110fveq2d 6846 . . . . . . . 8 (𝑥 = ∅ → (∏t‘(𝐹𝑥)) = (∏t‘∅))
1211eleq1d 2822 . . . . . . 7 (𝑥 = ∅ → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘∅) ∈ Comp))
1312imbi2d 340 . . . . . 6 (𝑥 = ∅ → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp)))
147, 13imbi12d 344 . . . . 5 (𝑥 = ∅ → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))))
15 sseq1 3969 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 reseq2 5932 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1716fveq2d 6846 . . . . . . . 8 (𝑥 = 𝑦 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝑦)))
1817eleq1d 2822 . . . . . . 7 (𝑥 = 𝑦 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝑦)) ∈ Comp))
1918imbi2d 340 . . . . . 6 (𝑥 = 𝑦 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
2015, 19imbi12d 344 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp))))
21 sseq1 3969 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
22 reseq2 5932 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
2322fveq2d 6846 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
2423eleq1d 2822 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
2524imbi2d 340 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
2621, 25imbi12d 344 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
27 sseq1 3969 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
28 reseq2 5932 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
2928fveq2d 6846 . . . . . . . 8 (𝑥 = 𝐴 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝐴)))
3029eleq1d 2822 . . . . . . 7 (𝑥 = 𝐴 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝐴)) ∈ Comp))
3130imbi2d 340 . . . . . 6 (𝑥 = 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
3227, 31imbi12d 344 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))))
33 0ex 5264 . . . . . . . . 9 ∅ ∈ V
34 f0 6723 . . . . . . . . 9 ∅:∅⟶Top
35 pttop 22933 . . . . . . . . 9 ((∅ ∈ V ∧ ∅:∅⟶Top) → (∏t‘∅) ∈ Top)
3633, 34, 35mp2an 690 . . . . . . . 8 (∏t‘∅) ∈ Top
37 eqid 2736 . . . . . . . . . . . . 13 (∏t‘∅) = (∏t‘∅)
3837ptuni 22945 . . . . . . . . . . . 12 ((∅ ∈ V ∧ ∅:∅⟶Top) → X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅))
3933, 34, 38mp2an 690 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅)
40 ixp0x 8864 . . . . . . . . . . . 12 X𝑥 ∈ ∅ (∅‘𝑥) = {∅}
41 snfi 8988 . . . . . . . . . . . 12 {∅} ∈ Fin
4240, 41eqeltri 2834 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) ∈ Fin
4339, 42eqeltrri 2835 . . . . . . . . . 10 (∏t‘∅) ∈ Fin
44 pwfi 9122 . . . . . . . . . 10 ( (∏t‘∅) ∈ Fin ↔ 𝒫 (∏t‘∅) ∈ Fin)
4543, 44mpbi 229 . . . . . . . . 9 𝒫 (∏t‘∅) ∈ Fin
46 pwuni 4906 . . . . . . . . 9 (∏t‘∅) ⊆ 𝒫 (∏t‘∅)
47 ssfi 9117 . . . . . . . . 9 ((𝒫 (∏t‘∅) ∈ Fin ∧ (∏t‘∅) ⊆ 𝒫 (∏t‘∅)) → (∏t‘∅) ∈ Fin)
4845, 46, 47mp2an 690 . . . . . . . 8 (∏t‘∅) ∈ Fin
4936, 48elini 4153 . . . . . . 7 (∏t‘∅) ∈ (Top ∩ Fin)
50 fincmp 22744 . . . . . . 7 ((∏t‘∅) ∈ (Top ∩ Fin) → (∏t‘∅) ∈ Comp)
5149, 50ax-mp 5 . . . . . 6 (∏t‘∅) ∈ Comp
52512a1i 12 . . . . 5 (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))
53 ssun1 4132 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
54 id 22 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
5553, 54sstrid 3955 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑦𝐴)
5655imim1i 63 . . . . . . 7 ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
57 eqid 2736 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘(𝐹𝑦))
58 eqid 2736 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘(𝐹 ↾ {𝑧}))
59 eqid 2736 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))
60 resabs1 5967 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦))
6153, 60ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦)
6261eqcomi 2745 . . . . . . . . . . . . . . 15 (𝐹𝑦) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)
6362fveq2i 6845 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦))
64 ssun2 4133 . . . . . . . . . . . . . . . . 17 {𝑧} ⊆ (𝑦 ∪ {𝑧})
65 resabs1 5967 . . . . . . . . . . . . . . . . 17 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧}))
6664, 65ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧})
6766eqcomi 2745 . . . . . . . . . . . . . . 15 (𝐹 ↾ {𝑧}) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧})
6867fveq2i 6845 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}))
69 eqid 2736 . . . . . . . . . . . . . 14 (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) = (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣))
70 vex 3449 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 vsnex 5386 . . . . . . . . . . . . . . . 16 {𝑧} ∈ V
7270, 71unex 7680 . . . . . . . . . . . . . . 15 (𝑦 ∪ {𝑧}) ∈ V
7372a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ∈ V)
74 simplr 767 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Comp)
75 cmptop 22746 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Comp → 𝑥 ∈ Top)
7675ssriv 3948 . . . . . . . . . . . . . . . 16 Comp ⊆ Top
77 fss 6685 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
7874, 76, 77sylancl 586 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Top)
79 simprr 771 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
8078, 79fssresd 6709 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ (𝑦 ∪ {𝑧})):(𝑦 ∪ {𝑧})⟶Top)
81 eqidd 2737 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
82 simprl 769 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧𝑦)
83 disjsn 4672 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
8482, 83sylibr 233 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∩ {𝑧}) = ∅)
8557, 58, 59, 63, 68, 69, 73, 80, 81, 84ptunhmeo 23159 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) ∈ (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧})))Homeo(∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))))
86 hmphi 23128 . . . . . . . . . . . . 13 ((𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) ∈ (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧})))Homeo(∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
8785, 86syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
881ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹 Fn 𝐴)
8964, 79sstrid 3955 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
90 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
9190snss 4746 . . . . . . . . . . . . . . . . 17 (𝑧𝐴 ↔ {𝑧} ⊆ 𝐴)
9289, 91sylibr 233 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧𝐴)
93 fnressn 7104 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑧𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9488, 92, 93syl2anc 584 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9594fveq2d 6846 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
96 eqid 2736 . . . . . . . . . . . . . . . . 17 (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩})
9790a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
9874, 92ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Comp)
9976, 98sselid 3942 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Top)
100 toptopon2 22267 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧) ∈ Top ↔ (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
10199, 100sylib 217 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
10296, 97, 101pt1hmeo 23157 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})))
103 hmphi 23128 . . . . . . . . . . . . . . . 16 ((𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
104102, 103syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
105 cmphmph 23139 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) → ((𝐹𝑧) ∈ Comp → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp))
106104, 98, 105sylc 65 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp)
10795, 106eqeltrd 2838 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp)
108 txcmp 22994 . . . . . . . . . . . . . 14 (((∏t‘(𝐹𝑦)) ∈ Comp ∧ (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp)
109108expcom 414 . . . . . . . . . . . . 13 ((∏t‘(𝐹 ↾ {𝑧})) ∈ Comp → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
111 cmphmph 23139 . . . . . . . . . . . 12 (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) → (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
11287, 110, 111sylsyld 61 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
113112expcom 414 . . . . . . . . . 10 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
114113a2d 29 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
115114ex 413 . . . . . . . 8 𝑧𝑦 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
116115a2d 29 . . . . . . 7 𝑧𝑦 → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
11756, 116syl5 34 . . . . . 6 𝑧𝑦 → ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
118117adantl 482 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
11914, 20, 26, 32, 52, 118findcard2s 9109 . . . 4 (𝐴 ∈ Fin → (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
1206, 119mpi 20 . . 3 (𝐴 ∈ Fin → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))
121120anabsi5 667 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)
1225, 121eqeltrrd 2839 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Xcixp 8835  Fincfn 8883  tcpt 17320  Topctop 22242  TopOnctopon 22259  Compccmp 22737   ×t ctx 22911  Homeochmeo 23104  chmph 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-hmph 23107
This theorem is referenced by:  poimirlem30  36108
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