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Theorem neiin 32311
Description: Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
Assertion
Ref Expression
neiin ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))

Proof of Theorem neiin
StepHypRef Expression
1 simpr 477 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2 simpl 473 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
43neiss2 20899 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 𝐽)
53neii1 20904 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 𝐽)
63neiint 20902 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑀 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
72, 4, 5, 6syl3anc 1325 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
81, 7mpbid 222 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9 ssinss1 3839 . . . . . 6 (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
108, 9syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
11103adant3 1080 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑀))
12 inss2 3832 . . . . 5 (𝐴𝐵) ⊆ 𝐵
13 simpr 477 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14 simpl 473 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
153neiss2 20899 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 𝐽)
163neii1 20904 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
173neiint 20902 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐵 𝐽𝑁 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1814, 15, 16, 17syl3anc 1325 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
1913, 18mpbid 222 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20193adant2 1079 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
2112, 20syl5ss 3612 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘𝑁))
2211, 21ssind 3835 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23 simp1 1060 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
2453adant3 1080 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 𝐽)
25163adant2 1079 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 𝐽)
263ntrin 20859 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 𝐽𝑁 𝐽) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2723, 24, 25, 26syl3anc 1325 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((int‘𝐽)‘(𝑀𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
2822, 27sseqtr4d 3640 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁)))
29 ssinss1 3839 . . . . 5 (𝐴 𝐽 → (𝐴𝐵) ⊆ 𝐽)
304, 29syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴𝐵) ⊆ 𝐽)
31 ssinss1 3839 . . . . 5 (𝑀 𝐽 → (𝑀𝑁) ⊆ 𝐽)
325, 31syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀𝑁) ⊆ 𝐽)
333neiint 20902 . . . 4 ((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝐽 ∧ (𝑀𝑁) ⊆ 𝐽) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
342, 30, 32, 33syl3anc 1325 . . 3 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
35343adant3 1080 . 2 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)) ↔ (𝐴𝐵) ⊆ ((int‘𝐽)‘(𝑀𝑁))))
3628, 35mpbird 247 1 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  cin 3571  wss 3572   cuni 4434  cfv 5886  Topctop 20692  intcnt 20815  neicnei 20895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-top 20693  df-cld 20817  df-ntr 20818  df-cls 20819  df-nei 20896
This theorem is referenced by: (None)
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