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Theorem neitx 21631
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 21131 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴𝑋)
32ad2ant2r 741 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴𝑋)
4 neitx.y . . . . . 6 𝑌 = 𝐾
54neii1 21131 . . . . 5 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵𝑌)
65ad2ant2l 740 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵𝑌)
7 xpss12 5264 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
83, 6, 7syl2anc 573 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
91, 4txuni 21616 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
109adantr 466 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
118, 10sseqtrd 3790 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾))
12 simp-5l 772 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
13 simp-4r 770 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐽)
14 simplr 752 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐾)
15 txopn 21626 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎𝐽𝑏𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
1612, 13, 14, 15syl12anc 1474 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
17 simpr1l 1290 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝐶𝑎)
18173anassrs 1453 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐶𝑎)
19 simprl 754 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐷𝑏)
20 xpss12 5264 . . . . . 6 ((𝐶𝑎𝐷𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
2118, 19, 20syl2anc 573 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
22 simpr1r 1292 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝑎𝐴)
23223anassrs 1453 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐴)
24 simprr 756 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐵)
25 xpss12 5264 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
2623, 24, 25syl2anc 573 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
27 sseq2 3776 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28 sseq1 3775 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
2927, 28anbi12d 616 . . . . . 6 (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
3029rspcev 3460 . . . . 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 1474 . . . 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 21133 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3332ad2ant2l 740 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3433ad2antrr 705 . . . 4 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3531, 34r19.29a 3226 . . 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 21133 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3736ad2ant2r 741 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3835, 37r19.29a 3226 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
39 txtop 21593 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4039adantr 466 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
411neiss2 21126 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶𝑋)
4241ad2ant2r 741 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶𝑋)
434neiss2 21126 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷𝑌)
4443ad2ant2l 740 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷𝑌)
45 xpss12 5264 . . . . 5 ((𝐶𝑋𝐷𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4642, 44, 45syl2anc 573 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4746, 10sseqtrd 3790 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝐽 ×t 𝐾))
48 eqid 2771 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
4948isnei 21128 . . 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 573 . 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 692 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 382   = wceq 1631  wcel 2145  wrex 3062  wss 3723   cuni 4574   × cxp 5247  cfv 6031  (class class class)co 6793  Topctop 20918  neicnei 21122   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-nei 21123  df-tx 21586
This theorem is referenced by:  utop2nei  22274  utop3cls  22275
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