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Theorem neitx 21739
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 21239 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴𝑋)
32ad2ant2r 754 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴𝑋)
4 neitx.y . . . . . 6 𝑌 = 𝐾
54neii1 21239 . . . . 5 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵𝑌)
65ad2ant2l 753 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵𝑌)
7 xpss12 5327 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
83, 6, 7syl2anc 580 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
91, 4txuni 21724 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
109adantr 473 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
118, 10sseqtrd 3837 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝐽 ×t 𝐾))
12 simp-5l 806 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
13 simp-4r 804 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐽)
14 simplr 786 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐾)
15 txopn 21734 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎𝐽𝑏𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
1612, 13, 14, 15syl12anc 866 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
17 simpr1l 1306 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝐶𝑎)
18173anassrs 1470 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐶𝑎)
19 simprl 788 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝐷𝑏)
20 xpss12 5327 . . . . . 6 ((𝐶𝑎𝐷𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
2118, 19, 20syl2anc 580 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
22 simpr1r 1308 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ ((𝐶𝑎𝑎𝐴) ∧ 𝑏𝐾 ∧ (𝐷𝑏𝑏𝐵))) → 𝑎𝐴)
23223anassrs 1470 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑎𝐴)
24 simprr 790 . . . . . 6 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → 𝑏𝐵)
25 xpss12 5327 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
2623, 24, 25syl2anc 580 . . . . 5 (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) ∧ 𝑏𝐾) ∧ (𝐷𝑏𝑏𝐵)) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
27 sseq2 3823 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28 sseq1 3822 . . . . . . 7 (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
2927, 28anbi12d 625 . . . . . 6 (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
3029rspcev 3497 . . . . 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 866 . . . 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 21241 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3332ad2ant2l 753 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3433ad2antrr 718 . . . 4 (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎𝐽) ∧ (𝐶𝑎𝑎𝐴)) → ∃𝑏𝐾 (𝐷𝑏𝑏𝐵))
3531, 34r19.29a 3259 . . 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 21241 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3736ad2ant2r 754 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎𝐽 (𝐶𝑎𝑎𝐴))
3835, 37r19.29a 3259 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐𝑐 ⊆ (𝐴 × 𝐵)))
39 txtop 21701 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4039adantr 473 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
411neiss2 21234 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶𝑋)
4241ad2ant2r 754 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶𝑋)
434neiss2 21234 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷𝑌)
4443ad2ant2l 753 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷𝑌)
45 xpss12 5327 . . . . 5 ((𝐶𝑋𝐷𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4642, 44, 45syl2anc 580 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
4746, 10sseqtrd 3837 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝐽 ×t 𝐾))
48 eqid 2799 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
4948isnei 21236 . . 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 580 . 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 705 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 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  wss 3769   cuni 4628   × cxp 5310  cfv 6101  (class class class)co 6878  Topctop 21026  neicnei 21230   ×t ctx 21692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-topgen 16419  df-top 21027  df-topon 21044  df-bases 21079  df-nei 21231  df-tx 21694
This theorem is referenced by:  utop2nei  22382  utop3cls  22383
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