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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcmp | Structured version Visualization version GIF version |
Description: If two atoms are comparable, they are equal. (atsseq 32376 analog.) (Contributed by NM, 13-Oct-2011.) |
Ref | Expression |
---|---|
atcmp.l | ⊢ ≤ = (le‘𝐾) |
atcmp.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atcmp | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlpos 39283 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | 3ad2ant1 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atcmp.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 39271 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1133 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 4 | atbase 39271 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
8 | 7 | 3ad2ant3 1134 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ (Base‘𝐾)) |
9 | eqid 2735 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
10 | 3, 9 | atl0cl 39285 | . . 3 ⊢ (𝐾 ∈ AtLat → (0.‘𝐾) ∈ (Base‘𝐾)) |
11 | 10 | 3ad2ant1 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾) ∈ (Base‘𝐾)) |
12 | eqid 2735 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 9, 12, 4 | atcvr0 39270 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
14 | 13 | 3adant3 1131 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
15 | 9, 12, 4 | atcvr0 39270 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
16 | 15 | 3adant2 1130 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
17 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
18 | 3, 17, 12 | cvrcmp 39265 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (0.‘𝐾) ∈ (Base‘𝐾)) ∧ ((0.‘𝐾)( ⋖ ‘𝐾)𝑃 ∧ (0.‘𝐾)( ⋖ ‘𝐾)𝑄)) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
19 | 2, 6, 8, 11, 14, 16, 18 | syl132anc 1387 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Posetcpo 18365 0.cp0 18481 ⋖ ccvr 39244 Atomscatm 39245 AtLatcal 39246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-proset 18352 df-poset 18371 df-plt 18388 df-glb 18405 df-p0 18483 df-lat 18490 df-covers 39248 df-ats 39249 df-atl 39280 |
This theorem is referenced by: atncmp 39294 atnlt 39295 atnle 39299 cvlsupr2 39325 cvratlem 39404 2atjm 39428 atbtwn 39429 2atm 39510 2llnmeqat 39554 dalem25 39681 dalem55 39710 dalem57 39712 snatpsubN 39733 pmapat 39746 2llnma1b 39769 cdlemblem 39776 lhp2at0nle 40018 lhpat3 40029 4atexlemcnd 40055 trlval3 40170 cdlemc5 40178 cdleme3 40220 cdleme7 40232 cdleme11k 40251 cdleme16b 40262 cdleme16e 40265 cdleme16f 40266 cdlemednpq 40282 cdleme20j 40301 cdleme22aa 40322 cdleme22cN 40325 cdleme22d 40326 cdlemf2 40545 cdlemb3 40589 cdlemg12e 40630 cdlemg17dALTN 40647 cdlemg19a 40666 cdlemg27b 40679 cdlemg31d 40683 trlcone 40711 cdlemi 40803 tendotr 40813 cdlemk17 40841 cdlemk52 40937 cdleml1N 40959 dia2dimlem1 41047 dia2dimlem2 41048 dia2dimlem3 41049 |
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