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Theorem ptpconn 34679
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 34671 . . . . 5 (𝑥 ∈ PConn → 𝑥 ∈ Top)
21ssriv 3978 . . . 4 PConn ⊆ Top
3 fss 6724 . . . 4 ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
42, 3mpan2 688 . . 3 (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5 pttop 23407 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
64, 5sylan2 592 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ Top)
7 fvi 6957 . . . . . . . . . 10 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
87ad2antrr 723 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ( I ‘𝐴) = 𝐴)
98eleq2d 2811 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡𝐴))
109biimpa 476 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡𝐴)
11 simplr 766 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝐹:𝐴⟶PConn)
1211ffvelcdmda 7076 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝐹𝑡) ∈ PConn)
13 simprl 768 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 (∏t𝐹))
14 eqid 2724 . . . . . . . . . . . . . . . 16 (∏t𝐹) = (∏t𝐹)
1514ptuni 23419 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
164, 15sylan2 592 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝐴⟶PConn) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1716adantr 480 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1813, 17eleqtrrd 2828 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥X𝑡𝐴 (𝐹𝑡))
19 vex 3470 . . . . . . . . . . . . 13 𝑥 ∈ V
2019elixp 8893 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡))
2322r19.21bi 3240 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑥𝑡) ∈ (𝐹𝑡))
24 simprr 770 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 (∏t𝐹))
2524, 17eleqtrrd 2828 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦X𝑡𝐴 (𝐹𝑡))
26 vex 3470 . . . . . . . . . . . . 13 𝑦 ∈ V
2726elixp 8893 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡))
3029r19.21bi 3240 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑦𝑡) ∈ (𝐹𝑡))
31 eqid 2724 . . . . . . . . . 10 (𝐹𝑡) = (𝐹𝑡)
3231pconncn 34670 . . . . . . . . 9 (((𝐹𝑡) ∈ PConn ∧ (𝑥𝑡) ∈ (𝐹𝑡) ∧ (𝑦𝑡) ∈ (𝐹𝑡)) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
3312, 23, 30, 32syl3anc 1368 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
34 df-rex 3063 . . . . . . . 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 590 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3736ralrimiva 3138 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
38 fvex 6894 . . . . . 6 ( I ‘𝐴) ∈ V
39 eleq1 2813 . . . . . . 7 (𝑓 = (𝑔𝑡) → (𝑓 ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑡) ∈ (II Cn (𝐹𝑡))))
40 fveq1 6880 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘0) = ((𝑔𝑡)‘0))
4140eqeq1d 2726 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘0) = (𝑥𝑡) ↔ ((𝑔𝑡)‘0) = (𝑥𝑡)))
42 fveq1 6880 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘1) = ((𝑔𝑡)‘1))
4342eqeq1d 2726 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘1) = (𝑦𝑡) ↔ ((𝑔𝑡)‘1) = (𝑦𝑡)))
4441, 43anbi12d 630 . . . . . . 7 (𝑓 = (𝑔𝑡) → (((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
4539, 44anbi12d 630 . . . . . 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 10476 . . . . 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 24720 . . . . . . 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 772 . . . . . 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 480 . . . . . . 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 480 . . . . . . . . . . . . 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 2811 . . . . . . . . . . . 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 477 . . . . . . . . . . 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 770 . . . . . . . . . . . 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 6881 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (𝑔𝑡) = (𝑔𝑖))
58 fveq2 6881 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝐹𝑡) = (𝐹𝑖))
5958oveq2d 7417 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (II Cn (𝐹𝑡)) = (II Cn (𝐹𝑖)))
6057, 59eleq12d 2819 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑖) ∈ (II Cn (𝐹𝑖))))
6157fveq1d 6883 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘0) = ((𝑔𝑖)‘0))
62 fveq2 6881 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑥𝑡) = (𝑥𝑖))
6361, 62eqeq12d 2740 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘0) = (𝑥𝑡) ↔ ((𝑔𝑖)‘0) = (𝑥𝑖)))
6457fveq1d 6883 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘1) = ((𝑔𝑖)‘1))
65 fveq2 6881 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑦𝑡) = (𝑦𝑖))
6664, 65eqeq12d 2740 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘1) = (𝑦𝑡) ↔ ((𝑔𝑖)‘1) = (𝑦𝑖)))
6763, 66anbi12d 630 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)) ↔ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
6860, 67anbi12d 630 . . . . . . . . . . . . 13 (𝑡 = 𝑖 → (((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ↔ ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))))
6968rspccva 3603 . . . . . . . . . . . 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 579 . . . . . . . . . . 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 590 . . . . . . . . . 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 494 . . . . . . . . 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 24722 . . . . . . . . . 10 (0[,]1) = II
74 eqid 2724 . . . . . . . . . 10 (𝐹𝑖) = (𝐹𝑖)
7573, 74cnf 23071 . . . . . . . . 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 6950 . . . . . . 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 2826 . . . . . 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 23452 . . . . 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 495 . . . . . . . 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 494 . . . . . . 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 5238 . . . . . 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 13442 . . . . . . 7 0 ∈ (0[,]1)
84 mptexg 7214 . . . . . . . 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 6881 . . . . . . . . 9 (𝑧 = 0 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘0))
8786mpteq2dv 5240 . . . . . . . 8 (𝑧 = 0 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
88 eqid 2724 . . . . . . . 8 (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))
8987, 88fvmptg 6986 . . . . . . 7 ((0 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9083, 85, 89sylancr 586 . . . . . 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 494 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 Fn 𝐴)
9291adantr 480 . . . . . . 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 6940 . . . . . . 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 2774 . . . . 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 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)‘1) = (𝑦𝑖))
9796mpteq2dva 5238 . . . . . 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 13443 . . . . . . 7 1 ∈ (0[,]1)
99 mptexg 7214 . . . . . . . 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 6881 . . . . . . . . 9 (𝑧 = 1 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘1))
102101mpteq2dv 5240 . . . . . . . 8 (𝑧 = 1 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
103102, 88fvmptg 6986 . . . . . . 7 ((1 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10498, 100, 103sylancr 586 . . . . . 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 494 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 Fn 𝐴)
106105adantr 480 . . . . . . 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 6940 . . . . . . 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 2774 . . . . 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 6880 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0))
111110eqeq1d 2726 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥))
112 fveq1 6880 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1))
113112eqeq1d 2726 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦))
114111, 113anbi12d 630 . . . . . 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 3604 . . . . 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 834 . . . 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 1930 . . 3 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118117ralrimivva 3192 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119 eqid 2724 . . 3 (∏t𝐹) = (∏t𝐹)
120119ispconn 34669 . 2 ((∏t𝐹) ∈ PConn ↔ ((∏t𝐹) ∈ Top ∧ ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
1216, 118, 120sylanbrc 582 1 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  wss 3940   cuni 4899  cmpt 5221   I cid 5563   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7401  Xcixp 8886  0cc0 11105  1c1 11106  [,]cicc 13323  tcpt 17382  Topctop 22716  TopOnctopon 22733   Cn ccn 23049  IIcii 24716  PConncpconn 34665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-reg 9582  ax-inf2 9631  ax-ac2 10453  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-map 8817  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fi 9401  df-sup 9432  df-inf 9433  df-r1 9754  df-rank 9755  df-card 9929  df-ac 10106  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-icc 13327  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-topgen 17387  df-pt 17388  df-psmet 21219  df-xmet 21220  df-met 21221  df-bl 21222  df-mopn 21223  df-top 22717  df-topon 22734  df-bases 22770  df-cn 23052  df-cnp 23053  df-ii 24718  df-pconn 34667
This theorem is referenced by: (None)
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