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Theorem ptpconn 33827
Description: The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)
Assertion
Ref Expression
ptpconn ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ PConn)

Proof of Theorem ptpconn
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑧 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pconntop 33819 . . . . 5 (𝑥 ∈ PConn → 𝑥 ∈ Top)
21ssriv 3948 . . . 4 PConn ⊆ Top
3 fss 6685 . . . 4 ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
42, 3mpan2 689 . . 3 (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5 pttop 22933 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
64, 5sylan2 593 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ Top)
7 fvi 6917 . . . . . . . . . 10 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
87ad2antrr 724 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ( I ‘𝐴) = 𝐴)
98eleq2d 2823 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡𝐴))
109biimpa 477 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡𝐴)
11 simplr 767 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝐹:𝐴⟶PConn)
1211ffvelcdmda 7035 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝐹𝑡) ∈ PConn)
13 simprl 769 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 (∏t𝐹))
14 eqid 2736 . . . . . . . . . . . . . . . 16 (∏t𝐹) = (∏t𝐹)
1514ptuni 22945 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
164, 15sylan2 593 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝐴⟶PConn) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1716adantr 481 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1813, 17eleqtrrd 2841 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥X𝑡𝐴 (𝐹𝑡))
19 vex 3449 . . . . . . . . . . . . 13 𝑥 ∈ V
2019elixp 8842 . . . . . . . . . . . 12 (𝑥X𝑡𝐴 (𝐹𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2118, 20sylib 217 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2221simprd 496 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡))
2322r19.21bi 3234 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑥𝑡) ∈ (𝐹𝑡))
24 simprr 771 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 (∏t𝐹))
2524, 17eleqtrrd 2841 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦X𝑡𝐴 (𝐹𝑡))
26 vex 3449 . . . . . . . . . . . . 13 𝑦 ∈ V
2726elixp 8842 . . . . . . . . . . . 12 (𝑦X𝑡𝐴 (𝐹𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2825, 27sylib 217 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2928simprd 496 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡))
3029r19.21bi 3234 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑦𝑡) ∈ (𝐹𝑡))
31 eqid 2736 . . . . . . . . . 10 (𝐹𝑡) = (𝐹𝑡)
3231pconncn 33818 . . . . . . . . 9 (((𝐹𝑡) ∈ PConn ∧ (𝑥𝑡) ∈ (𝐹𝑡) ∧ (𝑦𝑡) ∈ (𝐹𝑡)) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
3312, 23, 30, 32syl3anc 1371 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
34 df-rex 3074 . . . . . . . 8 (∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3533, 34sylib 217 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3610, 35syldan 591 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3736ralrimiva 3143 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
38 fvex 6855 . . . . . 6 ( I ‘𝐴) ∈ V
39 eleq1 2825 . . . . . . 7 (𝑓 = (𝑔𝑡) → (𝑓 ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑡) ∈ (II Cn (𝐹𝑡))))
40 fveq1 6841 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘0) = ((𝑔𝑡)‘0))
4140eqeq1d 2738 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘0) = (𝑥𝑡) ↔ ((𝑔𝑡)‘0) = (𝑥𝑡)))
42 fveq1 6841 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘1) = ((𝑔𝑡)‘1))
4342eqeq1d 2738 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘1) = (𝑦𝑡) ↔ ((𝑔𝑡)‘1) = (𝑦𝑡)))
4441, 43anbi12d 631 . . . . . . 7 (𝑓 = (𝑔𝑡) → (((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
4539, 44anbi12d 631 . . . . . 6 (𝑓 = (𝑔𝑡) → ((𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))) ↔ ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
4638, 45ac6s2 10422 . . . . 5 (∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
4737, 46syl 17 . . . 4 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
48 iitopon 24242 . . . . . . 7 II ∈ (TopOn‘(0[,]1))
4948a1i 11 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → II ∈ (TopOn‘(0[,]1)))
50 simplll 773 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐴𝑉)
5111adantr 481 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐹:𝐴⟶PConn)
5251, 4syl 17 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐹:𝐴⟶Top)
538adantr 481 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ( I ‘𝐴) = 𝐴)
5453eleq2d 2823 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖 ∈ ( I ‘𝐴) ↔ 𝑖𝐴))
5554biimpar 478 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → 𝑖 ∈ ( I ‘𝐴))
56 simprr 771 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
57 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (𝑔𝑡) = (𝑔𝑖))
58 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝐹𝑡) = (𝐹𝑖))
5958oveq2d 7373 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (II Cn (𝐹𝑡)) = (II Cn (𝐹𝑖)))
6057, 59eleq12d 2832 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑖) ∈ (II Cn (𝐹𝑖))))
6157fveq1d 6844 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘0) = ((𝑔𝑖)‘0))
62 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑥𝑡) = (𝑥𝑖))
6361, 62eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘0) = (𝑥𝑡) ↔ ((𝑔𝑖)‘0) = (𝑥𝑖)))
6457fveq1d 6844 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘1) = ((𝑔𝑖)‘1))
65 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑦𝑡) = (𝑦𝑖))
6664, 65eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘1) = (𝑦𝑡) ↔ ((𝑔𝑖)‘1) = (𝑦𝑖)))
6763, 66anbi12d 631 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)) ↔ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
6860, 67anbi12d 631 . . . . . . . . . . . . 13 (𝑡 = 𝑖 → (((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ↔ ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))))
6968rspccva 3580 . . . . . . . . . . . 12 ((∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7056, 69sylan 580 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7155, 70syldan 591 . . . . . . . . . 10 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7271simpld 495 . . . . . . . . 9 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖) ∈ (II Cn (𝐹𝑖)))
73 iiuni 24244 . . . . . . . . . 10 (0[,]1) = II
74 eqid 2736 . . . . . . . . . 10 (𝐹𝑖) = (𝐹𝑖)
7573, 74cnf 22597 . . . . . . . . 9 ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) → (𝑔𝑖):(0[,]1)⟶ (𝐹𝑖))
7672, 75syl 17 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖):(0[,]1)⟶ (𝐹𝑖))
7776feqmptd 6910 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖) = (𝑧 ∈ (0[,]1) ↦ ((𝑔𝑖)‘𝑧)))
7877, 72eqeltrrd 2839 . . . . . 6 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑧 ∈ (0[,]1) ↦ ((𝑔𝑖)‘𝑧)) ∈ (II Cn (𝐹𝑖)))
7914, 49, 50, 52, 78ptcn 22978 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) ∈ (II Cn (∏t𝐹)))
8071simprd 496 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))
8180simpld 495 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖)‘0) = (𝑥𝑖))
8281mpteq2dva 5205 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) = (𝑖𝐴 ↦ (𝑥𝑖)))
83 0elunit 13386 . . . . . . 7 0 ∈ (0[,]1)
84 mptexg 7171 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V)
8550, 84syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V)
86 fveq2 6842 . . . . . . . . 9 (𝑧 = 0 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘0))
8786mpteq2dv 5207 . . . . . . . 8 (𝑧 = 0 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
88 eqid 2736 . . . . . . . 8 (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))
8987, 88fvmptg 6946 . . . . . . 7 ((0 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9083, 85, 89sylancr 587 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9121simpld 495 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 Fn 𝐴)
9291adantr 481 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑥 Fn 𝐴)
93 dffn5 6901 . . . . . . 7 (𝑥 Fn 𝐴𝑥 = (𝑖𝐴 ↦ (𝑥𝑖)))
9492, 93sylib 217 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑥 = (𝑖𝐴 ↦ (𝑥𝑖)))
9582, 90, 943eqtr4d 2786 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥)
9680simprd 496 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖)‘1) = (𝑦𝑖))
9796mpteq2dva 5205 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) = (𝑖𝐴 ↦ (𝑦𝑖)))
98 1elunit 13387 . . . . . . 7 1 ∈ (0[,]1)
99 mptexg 7171 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V)
10050, 99syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V)
101 fveq2 6842 . . . . . . . . 9 (𝑧 = 1 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘1))
102101mpteq2dv 5207 . . . . . . . 8 (𝑧 = 1 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
103102, 88fvmptg 6946 . . . . . . 7 ((1 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10498, 100, 103sylancr 587 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10528simpld 495 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 Fn 𝐴)
106105adantr 481 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑦 Fn 𝐴)
107 dffn5 6901 . . . . . . 7 (𝑦 Fn 𝐴𝑦 = (𝑖𝐴 ↦ (𝑦𝑖)))
108106, 107sylib 217 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑦 = (𝑖𝐴 ↦ (𝑦𝑖)))
10997, 104, 1083eqtr4d 2786 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)
110 fveq1 6841 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0))
111110eqeq1d 2738 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥))
112 fveq1 6841 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1))
113112eqeq1d 2738 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦))
114111, 113anbi12d 631 . . . . . 6 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)))
115114rspcev 3581 . . . . 5 (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) ∈ (II Cn (∏t𝐹)) ∧ (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
11679, 95, 109, 115syl12anc 835 . . . 4 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
11747, 116exlimddv 1938 . . 3 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118117ralrimivva 3197 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119 eqid 2736 . . 3 (∏t𝐹) = (∏t𝐹)
120119ispconn 33817 . 2 ((∏t𝐹) ∈ PConn ↔ ((∏t𝐹) ∈ Top ∧ ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
1216, 118, 120sylanbrc 583 1 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910   cuni 4865  cmpt 5188   I cid 5530   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Xcixp 8835  0cc0 11051  1c1 11052  [,]cicc 13267  tcpt 17320  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  IIcii 24238  PConncpconn 33813
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  ax-reg 9528  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  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-se 5589  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-pred 6253  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-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-topgen 17325  df-pt 17326  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-ii 24240  df-pconn 33815
This theorem is referenced by: (None)
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