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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tosglb | Structured version Visualization version GIF version |
Description: Same theorem as toslub 31777, 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 31778 | . . . 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 7332 | . . 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 260 | . . . 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 18257 | . . 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 18307 | . . . . . . 7 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))) |
12 | 11 | ibi 266 | . . . . . 6 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))) |
13 | 12 | simpld 495 | . . . . 5 ⊢ (𝐾 ∈ Toset → < Or 𝐵) |
14 | cnvso 6240 | . . . . 5 ⊢ ( < Or 𝐵 ↔ ◡ < Or 𝐵) | |
15 | 13, 14 | sylib 217 | . . . 4 ⊢ (𝐾 ∈ Toset → ◡ < Or 𝐵) |
16 | id 22 | . . . . 5 ⊢ (◡ < Or 𝐵 → ◡ < Or 𝐵) | |
17 | 16 | supval2 9390 | . . . 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 9378 | . . . 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ⊆ wss 3910 class class class wbr 5105 I cid 5530 Or wor 5544 ◡ccnv 5632 ↾ cres 5635 ‘cfv 6496 ℩crio 7311 supcsup 9375 infcinf 9376 Basecbs 17082 lecple 17139 ltcplt 18196 glbcglb 18198 Tosetctos 18304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-sup 9377 df-inf 9378 df-proset 18183 df-poset 18201 df-plt 18218 df-glb 18235 df-toset 18305 |
This theorem is referenced by: xrsp0 31816 |
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