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Theorem neitx 23638
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 23137 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴𝑋)
32ad2ant2r 746 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴𝑋)
4 neitx.y . . . . . 6 𝑌 = 𝐾
54neii1 23137 . . . . 5 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵𝑌)
65ad2ant2l 745 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵𝑌)
7 xpss12 5715 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
83, 6, 7syl2anc 583 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
91, 4txuni 23623 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
109adantr 480 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
118, 10sseqtrd 4049 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾))
12 simp-5l 784 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
13 simp-4r 783 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐽)
14 simplr 768 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐾)
15 txopn 23633 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎𝐽𝑏𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
1612, 13, 14, 15syl12anc 836 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
17 simpr1l 1230 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝐶𝑎)
18173anassrs 1360 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐶𝑎)
19 simprl 770 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐷𝑏)
20 xpss12 5715 . . . . . 6 ((𝐶𝑎𝐷𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
2118, 19, 20syl2anc 583 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
22 simpr1r 1231 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝑎𝐴)
23223anassrs 1360 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐴)
24 simprr 772 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐵)
25 xpss12 5715 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
2623, 24, 25syl2anc 583 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
27 sseq2 4035 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28 sseq1 4034 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
2927, 28anbi12d 631 . . . . . 6 (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
3029rspcev 3635 . . . . 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 836 . . . 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 23139 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3332ad2ant2l 745 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3433ad2antrr 725 . . . 4 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3531, 34r19.29a 3168 . . 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 23139 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3736ad2ant2r 746 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3835, 37r19.29a 3168 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
39 txtop 23600 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4039adantr 480 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
411neiss2 23132 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶𝑋)
4241ad2ant2r 746 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶𝑋)
434neiss2 23132 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷𝑌)
4443ad2ant2l 745 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷𝑌)
45 xpss12 5715 . . . . 5 ((𝐶𝑋𝐷𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4642, 44, 45syl2anc 583 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4746, 10sseqtrd 4049 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝐽 ×t 𝐾))
48 eqid 2740 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
4948isnei 23134 . . 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 583 . 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 712 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 206  wa 395   = wceq 1537  wcel 2108  wrex 3076  wss 3976   cuni 4931   × cxp 5698  cfv 6575  (class class class)co 7450  Topctop 22922  neicnei 23128   ×t ctx 23591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-topgen 17505  df-top 22923  df-topon 22940  df-bases 22976  df-nei 23129  df-tx 23593
This theorem is referenced by:  utop2nei  24282  utop3cls  24283
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