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Theorem neitx 21350
Description: The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
Hypotheses
Ref Expression
neitx.x 𝑋 = 𝐽
neitx.y 𝑌 = 𝐾
Assertion
Ref Expression
neitx (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))

Proof of Theorem neitx
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neitx.x . . . . . 6 𝑋 = 𝐽
21neii1 20850 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴𝑋)
32ad2ant2r 782 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴𝑋)
4 neitx.y . . . . . 6 𝑌 = 𝐾
54neii1 20850 . . . . 5 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵𝑌)
65ad2ant2l 781 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵𝑌)
7 xpss12 5196 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
83, 6, 7syl2anc 692 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
91, 4txuni 21335 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
109adantr 481 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
118, 10sseqtrd 3626 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾))
12 simp-5l 807 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
13 simp-4r 806 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐽)
14 simplr 791 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐾)
15 txopn 21345 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎𝐽𝑏𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
1612, 13, 14, 15syl12anc 1321 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
17 simpr1l 1116 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝐶𝑎)
18173anassrs 1287 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐶𝑎)
19 simprl 793 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐷𝑏)
20 xpss12 5196 . . . . . 6 ((𝐶𝑎𝐷𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
2118, 19, 20syl2anc 692 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
22 simpr1r 1117 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝑎𝐴)
23223anassrs 1287 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐴)
24 simprr 795 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐵)
25 xpss12 5196 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
2623, 24, 25syl2anc 692 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
27 sseq2 3612 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28 sseq1 3611 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
2927, 28anbi12d 746 . . . . . 6 (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
3029rspcev 3299 . . . . 5 (((𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾) ∧ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
3116, 21, 26, 30syl12anc 1321 . . . 4 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
32 neii2 20852 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3332ad2ant2l 781 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3433ad2antrr 761 . . . 4 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3531, 34r19.29a 3073 . . 3 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
36 neii2 20852 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3736ad2ant2r 782 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3835, 37r19.29a 3073 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
39 txtop 21312 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4039adantr 481 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
411neiss2 20845 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶𝑋)
4241ad2ant2r 782 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶𝑋)
434neiss2 20845 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷𝑌)
4443ad2ant2l 781 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷𝑌)
45 xpss12 5196 . . . . 5 ((𝐶𝑋𝐷𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4642, 44, 45syl2anc 692 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4746, 10sseqtrd 3626 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝐽 ×t 𝐾))
48 eqid 2621 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
4948isnei 20847 . . 3 (((𝐽 ×t 𝐾) ∈ Top ∧ (𝐶 × 𝐷) ⊆ (𝐽 ×t 𝐾)) → ((𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)) ↔ ((𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾) ∧ ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))))
5040, 47, 49syl2anc 692 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ((𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)) ↔ ((𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾) ∧ ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))))
5111, 38, 50mpbir2and 956 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560   cuni 4409   × cxp 5082  cfv 5857  (class class class)co 6615  Topctop 20638  neicnei 20841   ×t ctx 21303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-nei 20842  df-tx 21305
This theorem is referenced by:  utop2nei  21994  utop3cls  21995
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