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Theorem ptcmpfi 21664
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 6083 . . . . 5 (𝐹:𝐴⟶Comp → 𝐹 Fn 𝐴)
2 fnresdm 6038 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
31, 2syl 17 . . . 4 (𝐹:𝐴⟶Comp → (𝐹𝐴) = 𝐹)
43adantl 481 . . 3 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (𝐹𝐴) = 𝐹)
54fveq2d 6233 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) = (∏t𝐹))
6 ssid 3657 . . . 4 𝐴𝐴
7 sseq1 3659 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
8 reseq2 5423 . . . . . . . . . 10 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
9 res0 5432 . . . . . . . . . 10 (𝐹 ↾ ∅) = ∅
108, 9syl6eq 2701 . . . . . . . . 9 (𝑥 = ∅ → (𝐹𝑥) = ∅)
1110fveq2d 6233 . . . . . . . 8 (𝑥 = ∅ → (∏t‘(𝐹𝑥)) = (∏t‘∅))
1211eleq1d 2715 . . . . . . 7 (𝑥 = ∅ → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘∅) ∈ Comp))
1312imbi2d 329 . . . . . 6 (𝑥 = ∅ → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp)))
147, 13imbi12d 333 . . . . 5 (𝑥 = ∅ → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))))
15 sseq1 3659 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 reseq2 5423 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1716fveq2d 6233 . . . . . . . 8 (𝑥 = 𝑦 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝑦)))
1817eleq1d 2715 . . . . . . 7 (𝑥 = 𝑦 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝑦)) ∈ Comp))
1918imbi2d 329 . . . . . 6 (𝑥 = 𝑦 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
2015, 19imbi12d 333 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp))))
21 sseq1 3659 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
22 reseq2 5423 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
2322fveq2d 6233 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
2423eleq1d 2715 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
2524imbi2d 329 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
2621, 25imbi12d 333 . . . . 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 3659 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
28 reseq2 5423 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
2928fveq2d 6233 . . . . . . . 8 (𝑥 = 𝐴 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝐴)))
3029eleq1d 2715 . . . . . . 7 (𝑥 = 𝐴 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝐴)) ∈ Comp))
3130imbi2d 329 . . . . . 6 (𝑥 = 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
3227, 31imbi12d 333 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))))
33 0ex 4823 . . . . . . . . 9 ∅ ∈ V
34 f0 6124 . . . . . . . . 9 ∅:∅⟶Top
35 pttop 21433 . . . . . . . . 9 ((∅ ∈ V ∧ ∅:∅⟶Top) → (∏t‘∅) ∈ Top)
3633, 34, 35mp2an 708 . . . . . . . 8 (∏t‘∅) ∈ Top
37 eqid 2651 . . . . . . . . . . . . 13 (∏t‘∅) = (∏t‘∅)
3837ptuni 21445 . . . . . . . . . . . 12 ((∅ ∈ V ∧ ∅:∅⟶Top) → X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅))
3933, 34, 38mp2an 708 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅)
40 ixp0x 7978 . . . . . . . . . . . 12 X𝑥 ∈ ∅ (∅‘𝑥) = {∅}
41 snfi 8079 . . . . . . . . . . . 12 {∅} ∈ Fin
4240, 41eqeltri 2726 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) ∈ Fin
4339, 42eqeltrri 2727 . . . . . . . . . 10 (∏t‘∅) ∈ Fin
44 pwfi 8302 . . . . . . . . . 10 ( (∏t‘∅) ∈ Fin ↔ 𝒫 (∏t‘∅) ∈ Fin)
4543, 44mpbi 220 . . . . . . . . 9 𝒫 (∏t‘∅) ∈ Fin
46 pwuni 4506 . . . . . . . . 9 (∏t‘∅) ⊆ 𝒫 (∏t‘∅)
47 ssfi 8221 . . . . . . . . 9 ((𝒫 (∏t‘∅) ∈ Fin ∧ (∏t‘∅) ⊆ 𝒫 (∏t‘∅)) → (∏t‘∅) ∈ Fin)
4845, 46, 47mp2an 708 . . . . . . . 8 (∏t‘∅) ∈ Fin
49 elin 3829 . . . . . . . 8 ((∏t‘∅) ∈ (Top ∩ Fin) ↔ ((∏t‘∅) ∈ Top ∧ (∏t‘∅) ∈ Fin))
5036, 48, 49mpbir2an 975 . . . . . . 7 (∏t‘∅) ∈ (Top ∩ Fin)
51 fincmp 21244 . . . . . . 7 ((∏t‘∅) ∈ (Top ∩ Fin) → (∏t‘∅) ∈ Comp)
5250, 51ax-mp 5 . . . . . 6 (∏t‘∅) ∈ Comp
53522a1i 12 . . . . 5 (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))
54 ssun1 3809 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
55 id 22 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
5654, 55syl5ss 3647 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑦𝐴)
5756imim1i 63 . . . . . . 7 ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
58 eqid 2651 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘(𝐹𝑦))
59 eqid 2651 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘(𝐹 ↾ {𝑧}))
60 eqid 2651 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))
61 resabs1 5462 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦))
6254, 61ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦)
6362eqcomi 2660 . . . . . . . . . . . . . . 15 (𝐹𝑦) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)
6463fveq2i 6232 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦))
65 ssun2 3810 . . . . . . . . . . . . . . . . 17 {𝑧} ⊆ (𝑦 ∪ {𝑧})
66 resabs1 5462 . . . . . . . . . . . . . . . . 17 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧}))
6765, 66ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧})
6867eqcomi 2660 . . . . . . . . . . . . . . 15 (𝐹 ↾ {𝑧}) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧})
6968fveq2i 6232 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}))
70 eqid 2651 . . . . . . . . . . . . . 14 (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) = (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣))
71 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
72 snex 4938 . . . . . . . . . . . . . . . 16 {𝑧} ∈ V
7371, 72unex 6998 . . . . . . . . . . . . . . 15 (𝑦 ∪ {𝑧}) ∈ V
7473a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ∈ V)
75 simplr 807 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Comp)
76 cmptop 21246 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Comp → 𝑥 ∈ Top)
7776ssriv 3640 . . . . . . . . . . . . . . . 16 Comp ⊆ Top
78 fss 6094 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
7975, 77, 78sylancl 695 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Top)
80 simprr 811 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
8179, 80fssresd 6109 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ (𝑦 ∪ {𝑧})):(𝑦 ∪ {𝑧})⟶Top)
82 eqidd 2652 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
83 simprl 809 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧𝑦)
84 disjsn 4278 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
8583, 84sylibr 224 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∩ {𝑧}) = ∅)
8658, 59, 60, 64, 69, 70, 74, 81, 82, 85ptunhmeo 21659 . . . . . . . . . . . . 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})))))
87 hmphi 21628 . . . . . . . . . . . . 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}))))
8886, 87syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
891ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹 Fn 𝐴)
9065, 80syl5ss 3647 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
91 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
9291snss 4348 . . . . . . . . . . . . . . . . 17 (𝑧𝐴 ↔ {𝑧} ⊆ 𝐴)
9390, 92sylibr 224 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧𝐴)
94 fnressn 6465 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑧𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9589, 93, 94syl2anc 694 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9695fveq2d 6233 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
97 eqid 2651 . . . . . . . . . . . . . . . . 17 (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩})
9891a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
9975, 93ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Comp)
10077, 99sseldi 3634 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Top)
101 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑧) = (𝐹𝑧)
102101toptopon 20770 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧) ∈ Top ↔ (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
103100, 102sylib 208 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
10497, 98, 103pt1hmeo 21657 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})))
105 hmphi 21628 . . . . . . . . . . . . . . . 16 ((𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
106104, 105syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
107 cmphmph 21639 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) → ((𝐹𝑧) ∈ Comp → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp))
108106, 99, 107sylc 65 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp)
10996, 108eqeltrd 2730 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp)
110 txcmp 21494 . . . . . . . . . . . . . 14 (((∏t‘(𝐹𝑦)) ∈ Comp ∧ (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp)
111110expcom 450 . . . . . . . . . . . . 13 ((∏t‘(𝐹 ↾ {𝑧})) ∈ Comp → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
112109, 111syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
113 cmphmph 21639 . . . . . . . . . . . 12 (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) → (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
11488, 112, 113sylsyld 61 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
115114expcom 450 . . . . . . . . . 10 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
116115a2d 29 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
117116ex 449 . . . . . . . 8 𝑧𝑦 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
118117a2d 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))))
11957, 118syl5 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))))
120119adantl 481 . . . . 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))))
12114, 20, 26, 32, 53, 120findcard2s 8242 . . . 4 (𝐴 ∈ Fin → (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
1226, 121mpi 20 . . 3 (𝐴 ∈ Fin → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))
123122anabsi5 875 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)
1245, 123eqeltrrd 2731 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762  cres 5145   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  Xcixp 7950  Fincfn 7997  tcpt 16146  Topctop 20746  TopOnctopon 20763  Compccmp 21237   ×t ctx 21411  Homeochmeo 21604  chmph 21605
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
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-ral 2946  df-rex 2947  df-reu 2948  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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  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-fi 8358  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-hmph 21607
This theorem is referenced by:  poimirlem30  33569
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