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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tosglb | Structured version Visualization version GIF version | ||
| Description: Same theorem as toslub 32964, 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 2736 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 6 | 1, 2, 3, 4, 5 | tosglblem 32965 | . . . 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 7408 | . . 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 2736 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 9 | biid 261 | . . . 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 18415 | . . 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 18465 | . . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))) | 
| 12 | 11 | ibi 267 | . . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))) | 
| 13 | 12 | simpld 494 | . . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵) | 
| 14 | cnvso 6307 | . . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵) | |
| 15 | 13, 14 | sylib 218 | . . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵) | 
| 16 | id 22 | . . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵) | |
| 17 | 16 | supval2 9496 | . . . 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 18 | . . 3 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) | 
| 19 | 7, 10, 18 | 3eqtr4d 2786 | . 2 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, ◡ < )) | 
| 20 | df-inf 9484 | . . . 4 ⊢ inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, ◡ < ) | |
| 21 | 20 | eqcomi 2745 | . . 3 ⊢ sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < ) | 
| 22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡ < ) = inf(𝐴, 𝐵, < )) | 
| 23 | 19, 22 | eqtrd 2776 | 1 ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5142 I cid 5576 Or wor 5590 ◡ccnv 5683 ↾ cres 5686 ‘cfv 6560 ℩crio 7388 supcsup 9481 infcinf 9482 Basecbs 17248 lecple 17305 ltcplt 18355 glbcglb 18357 Tosetctos 18462 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-sup 9483 df-inf 9484 df-proset 18341 df-poset 18360 df-plt 18376 df-glb 18393 df-toset 18463 | 
| This theorem is referenced by: xrsp0 33015 | 
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