MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trcfilu Structured version   Visualization version   GIF version

Theorem trcfilu 23446
Description: Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
trcfilu ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))

Proof of Theorem trcfilu
Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1135 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2 simp2l 1198 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (CauFilu𝑈))
3 iscfilu 23440 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
43biimpa 477 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
51, 2, 4syl2anc 584 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
65simpld 495 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (fBas‘𝑋))
7 simp3 1137 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐴𝑋)
8 simp2r 1199 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ¬ ∅ ∈ (𝐹t 𝐴))
9 trfbas2 22994 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))
109biimpar 478 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
116, 7, 8, 10syl21anc 835 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
122ad5antr 731 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐹 ∈ (CauFilu𝑈))
131adantr 481 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
1413elfvexd 6808 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑋 ∈ V)
157adantr 481 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴𝑋)
1614, 15ssexd 5248 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴 ∈ V)
1716ad4antr 729 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐴 ∈ V)
18 simplr 766 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑎𝐹)
19 elrestr 17139 . . . . . . 7 ((𝐹 ∈ (CauFilu𝑈) ∧ 𝐴 ∈ V ∧ 𝑎𝐹) → (𝑎𝐴) ∈ (𝐹t 𝐴))
2012, 17, 18, 19syl3anc 1370 . . . . . 6 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎𝐴) ∈ (𝐹t 𝐴))
21 inxp 5741 . . . . . . 7 ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎𝐴) × (𝑎𝐴))
22 simpr 485 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
2322ssrind 4169 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
24 simpllr 773 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
2523, 24sseqtrrd 3962 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ 𝑤)
2621, 25eqsstrrid 3970 . . . . . 6 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤)
27 id 22 . . . . . . . . 9 (𝑏 = (𝑎𝐴) → 𝑏 = (𝑎𝐴))
2827sqxpeqd 5621 . . . . . . . 8 (𝑏 = (𝑎𝐴) → (𝑏 × 𝑏) = ((𝑎𝐴) × (𝑎𝐴)))
2928sseq1d 3952 . . . . . . 7 (𝑏 = (𝑎𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤))
3029rspcev 3561 . . . . . 6 (((𝑎𝐴) ∈ (𝐹t 𝐴) ∧ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3120, 26, 30syl2anc 584 . . . . 5 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
325simprd 496 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3332r19.21bi 3134 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3433ad4ant13 748 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3531, 34r19.29a 3218 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3616, 16xpexd 7601 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37 simpr 485 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
38 elrest 17138 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
3938biimpa 477 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4013, 36, 37, 39syl21anc 835 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4135, 40r19.29a 3218 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
4241ralrimiva 3103 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
43 trust 23381 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
441, 7, 43syl2anc 584 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45 iscfilu 23440 . . 3 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → ((𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ ((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
4644, 45syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ ((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
4711, 42, 46mpbir2and 710 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  cin 3886  wss 3887  c0 4256   × cxp 5587  cfv 6433  (class class class)co 7275  t crest 17131  fBascfbas 20585  UnifOncust 23351  CauFiluccfilu 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-rest 17133  df-fbas 20594  df-ust 23352  df-cfilu 23439
This theorem is referenced by:  ucnextcn  23456
  Copyright terms: Public domain W3C validator