Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ordtrest2NEW Structured version   Visualization version   GIF version

Theorem ordtrest2NEW 29297
 Description: An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
ordtrest2NEW.2 (𝜑𝐾 ∈ Toset)
ordtrest2NEW.3 (𝜑𝐴𝐵)
ordtrest2NEW.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
Assertion
Ref Expression
ordtrest2NEW (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦   𝑥,𝐴,𝑦,𝑧   𝑧,   𝑧,𝐴   𝑧,𝐵   𝜑,𝑥,𝑦,𝑧   𝑧,𝐾

Proof of Theorem ordtrest2NEW
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtrest2NEW.2 . . . 4 (𝜑𝐾 ∈ Toset)
2 tospos 28989 . . . 4 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3 posprs 16772 . . . 4 (𝐾 ∈ Poset → 𝐾 ∈ Preset )
41, 2, 33syl 18 . . 3 (𝜑𝐾 ∈ Preset )
5 ordtrest2NEW.3 . . 3 (𝜑𝐴𝐵)
6 ordtNEW.b . . . 4 𝐵 = (Base‘𝐾)
7 ordtNEW.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
86, 7ordtrestNEW 29295 . . 3 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
94, 5, 8syl2anc 691 . 2 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
10 eqid 2610 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
11 eqid 2610 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})
126, 7, 10, 11ordtprsval 29292 . . . . . . 7 (𝐾 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))))
134, 12syl 17 . . . . . 6 (𝜑 → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))))
1413oveq1d 6564 . . . . 5 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
15 fibas 20592 . . . . . 6 (fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases
16 fvex 6113 . . . . . . . . 9 (Base‘𝐾) ∈ V
176, 16eqeltri 2684 . . . . . . . 8 𝐵 ∈ V
1817a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
1918, 5ssexd 4733 . . . . . 6 (𝜑𝐴 ∈ V)
20 tgrest 20773 . . . . . 6 (((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
2115, 19, 20sylancr 694 . . . . 5 (𝜑 → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
2214, 21eqtr4d 2647 . . . 4 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)))
23 firest 15916 . . . . 5 (fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴)) = ((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)
2423fveq2i 6106 . . . 4 (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴))
2522, 24syl6eqr 2662 . . 3 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))))
26 fvex 6113 . . . . . . . 8 (le‘𝐾) ∈ V
2726inex1 4727 . . . . . . 7 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
287, 27eqeltri 2684 . . . . . 6 ∈ V
2928inex1 4727 . . . . 5 ( ∩ (𝐴 × 𝐴)) ∈ V
30 ordttop 20814 . . . . 5 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
3129, 30mp1i 13 . . . 4 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
326, 7, 10, 11ordtprsuni 29293 . . . . . . . . 9 (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
334, 32syl 17 . . . . . . . 8 (𝜑𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
3433, 18eqeltrrd 2689 . . . . . . 7 (𝜑 ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
35 uniexb 6866 . . . . . . 7 (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V ↔ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
3634, 35sylibr 223 . . . . . 6 (𝜑 → ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
37 restval 15910 . . . . . 6 ((({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)))
3836, 19, 37syl2anc 691 . . . . 5 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)))
39 sseqin2 3779 . . . . . . . . . . . 12 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
405, 39sylib 207 . . . . . . . . . . 11 (𝜑 → (𝐵𝐴) = 𝐴)
41 eqid 2610 . . . . . . . . . . . . . . 15 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
4241ordttopon 20807 . . . . . . . . . . . . . 14 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
4329, 42mp1i 13 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
446, 7prsssdm 29291 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
454, 5, 44syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
4645fveq2d 6107 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom ( ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4743, 46eleqtrd 2690 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
48 toponmax 20543 . . . . . . . . . . . 12 ((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4947, 48syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
5040, 49eqeltrd 2688 . . . . . . . . . 10 (𝜑 → (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
51 elsni 4142 . . . . . . . . . . . 12 (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
5251ineq1d 3775 . . . . . . . . . . 11 (𝑣 ∈ {𝐵} → (𝑣𝐴) = (𝐵𝐴))
5352eleq1d 2672 . . . . . . . . . 10 (𝑣 ∈ {𝐵} → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5450, 53syl5ibrcom 236 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐵} → (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5554ralrimiv 2948 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
56 ordtrest2NEW.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
576, 7, 1, 5, 56ordtrest2NEWlem 29296 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
58 eqid 2610 . . . . . . . . . . . 12 (ODual‘𝐾) = (ODual‘𝐾)
5958, 6odubas 16956 . . . . . . . . . . 11 𝐵 = (Base‘(ODual‘𝐾))
607cnveqi 5219 . . . . . . . . . . . 12 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
61 cnvin 5459 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
62 cnvxp 5470 . . . . . . . . . . . . . 14 (𝐵 × 𝐵) = (𝐵 × 𝐵)
6362ineq2i 3773 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
64 eqid 2610 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
6558, 64oduleval 16954 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘(ODual‘𝐾))
6665ineq1i 3772 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6761, 63, 663eqtri 2636 . . . . . . . . . . . 12 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6860, 67eqtri 2632 . . . . . . . . . . 11 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6958odutos 28994 . . . . . . . . . . . 12 (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
701, 69syl 17 . . . . . . . . . . 11 (𝜑 → (ODual‘𝐾) ∈ Toset)
71 vex 3176 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
72 vex 3176 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
7371, 72brcnv 5227 . . . . . . . . . . . . . . . 16 (𝑦 𝑧𝑧 𝑦)
74 vex 3176 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
7572, 74brcnv 5227 . . . . . . . . . . . . . . . 16 (𝑧 𝑥𝑥 𝑧)
7673, 75anbi12ci 730 . . . . . . . . . . . . . . 15 ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦))
7776a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦)))
7877rabbiia 3161 . . . . . . . . . . . . 13 {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} = {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)}
7978, 56syl5eqss 3612 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
8079ancom2s 840 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
8159, 68, 70, 5, 80ordtrest2NEWlem 29296 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
82 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
8382, 72brcnv 5227 . . . . . . . . . . . . . . . . 17 (𝑤 𝑧𝑧 𝑤)
8483bicomi 213 . . . . . . . . . . . . . . . 16 (𝑧 𝑤𝑤 𝑧)
8584a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 𝑤𝑤 𝑧))
8685notbid 307 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧 𝑤 ↔ ¬ 𝑤 𝑧))
8786rabbidv 3164 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝐵 ∣ ¬ 𝑧 𝑤} = {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
8887mpteq2dv 4673 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
8988rneqd 5274 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
90 cnvin 5459 . . . . . . . . . . . . . . 15 ( ∩ (𝐴 × 𝐴)) = ( (𝐴 × 𝐴))
91 cnvxp 5470 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
9291ineq2i 3773 . . . . . . . . . . . . . . 15 ( (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9390, 92eqtri 2632 . . . . . . . . . . . . . 14 ( ∩ (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9493fveq2i 6106 . . . . . . . . . . . . 13 (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴)))
956ressprs 28986 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
964, 5, 95syl2anc 691 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾s 𝐴) ∈ Preset )
97 eqid 2610 . . . . . . . . . . . . . . . 16 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
98 eqid 2610 . . . . . . . . . . . . . . . 16 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
9997, 98ordtcnvNEW 29294 . . . . . . . . . . . . . . 15 ((𝐾s 𝐴) ∈ Preset → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
10096, 99syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
1016, 7prsss 29290 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
1024, 5, 101syl2anc 691 . . . . . . . . . . . . . . . . 17 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
103 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (𝐾s 𝐴) = (𝐾s 𝐴)
104103, 64ressle 15882 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
10519, 104syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
106103, 6ressbas2 15758 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
1075, 106syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = (Base‘(𝐾s 𝐴)))
108107sqxpeqd 5065 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
109105, 108ineq12d 3777 . . . . . . . . . . . . . . . . 17 (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
110102, 109eqtrd 2644 . . . . . . . . . . . . . . . 16 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
111110cnveqd 5220 . . . . . . . . . . . . . . 15 (𝜑( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
112111fveq2d 6107 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
113110fveq2d 6107 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
114100, 112, 1133eqtr4d 2654 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
11594, 114syl5reqr 2659 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
116115eleq2d 2673 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11789, 116raleqbidv 3129 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11881, 117mpbird 246 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
119 ralunb 3756 . . . . . . . . 9 (∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
12057, 118, 119sylanbrc 695 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
121 ralunb 3756 . . . . . . . 8 (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
12255, 120, 121sylanbrc 695 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
123 eqid 2610 . . . . . . . 8 (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴))
124123fmpt 6289 . . . . . . 7 (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))))
125122, 124sylib 207 . . . . . 6 (𝜑 → (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))))
126 frn 5966 . . . . . 6 ((𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))) → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
127125, 126syl 17 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12838, 127eqsstrd 3602 . . . 4 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
129 tgfiss 20606 . . . 4 (((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
13031, 128, 129syl2anc 691 . . 3 (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
13125, 130eqsstrd 3602 . 2 (𝜑 → ((ordTop‘ ) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
1329, 131eqssd 3585 1 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ficfi 8199  Basecbs 15695   ↾s cress 15696  lecple 15775   ↾t crest 15904  topGenctg 15921  ordTopcordt 15982   Preset cpreset 16749  Posetcpo 16763  Tosetctos 16856  ODualcodu 16951  Topctop 20517  TopOnctopon 20518  TopBasesctb 20520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-ple 15788  df-rest 15906  df-topgen 15927  df-ordt 15984  df-preset 16751  df-poset 16769  df-toset 16857  df-odu 16952  df-top 20521  df-bases 20522  df-topon 20523 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator