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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tosglb | Structured version Visualization version GIF version | ||
| Description: Same theorem as toslub 33206, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
| Ref | Expression |
|---|---|
| tosglb.b | ⊢ 𝐵 = (Base‘𝐾) |
| tosglb.l | ⊢ < = (lt‘𝐾) |
| tosglb.1 | ⊢ (𝜑 → 𝐾 ∈ Toset) |
| tosglb.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| Ref | Expression |
|---|---|
| tosglb | ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tosglb.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | tosglb.l | . . . . 5 ⊢ < = (lt‘𝐾) | |
| 3 | tosglb.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Toset) | |
| 4 | tosglb.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 5 | eqid 2765 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 6 | 1, 2, 3, 4, 5 | tosglblem 33207 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) |
| 7 | 6 | riotabidva 7376 | . . 3 ⊢ (𝜑 → (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎))) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) |
| 8 | eqid 2765 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 9 | biid 264 | . . . 4 ⊢ ((∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎))) | |
| 10 | 1, 5, 8, 9, 3, 4 | glbval 18413 | . . 3 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐(le‘𝐾)𝑏 → 𝑐(le‘𝐾)𝑎)))) |
| 11 | 1, 5, 2 | tosso 18463 | . . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))) |
| 12 | 11 | ibi 270 | . . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))) |
| 13 | 12 | simpld 499 | . . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵) |
| 14 | cnvso 6279 | . . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵) | |
| 15 | 13, 14 | sylib 221 | . . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵) |
| 16 | id 23 | . . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵) | |
| 17 | 16 | supval2 9403 | . . . 4 ⊢ (◡ < Or 𝐵 → sup(𝐴, 𝐵, ◡ < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) |
| 18 | 3, 15, 17 | 3syl 19 | . . 3 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) |
| 19 | 7, 10, 18 | 3eqtr4d 2810 | . 2 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, ◡ < )) |
| 20 | df-inf 9391 | . . . 4 ⊢ inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, ◡ < ) | |
| 21 | 20 | eqcomi 2774 | . . 3 ⊢ sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < ) |
| 22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < )) |
| 23 | 19, 22 | eqtrd 2800 | 1 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < )) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 class class class wbr 5105 I cid 5546 Or wor 5559 ◡ccnv 5651 ↾ cres 5654 ‘cfv 6525 ℩crio 7356 supcsup 9388 infcinf 9389 Basecbs 17259 lecple 17307 ltcplt 18354 glbcglb 18356 Tosetctos 18460 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-po 5560 df-so 5561 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-sup 9390 df-inf 9391 df-proset 18340 df-poset 18359 df-plt 18374 df-glb 18391 df-toset 18461 |
| This theorem is referenced by: xrsp0 33245 |
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