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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 32280 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 38999 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | 3ad2ant1 1130 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2726 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atcmp.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 38987 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1131 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 4 | atbase 38987 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
8 | 7 | 3ad2ant3 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ (Base‘𝐾)) |
9 | eqid 2726 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
10 | 3, 9 | atl0cl 39001 | . . 3 ⊢ (𝐾 ∈ AtLat → (0.‘𝐾) ∈ (Base‘𝐾)) |
11 | 10 | 3ad2ant1 1130 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾) ∈ (Base‘𝐾)) |
12 | eqid 2726 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 9, 12, 4 | atcvr0 38986 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
14 | 13 | 3adant3 1129 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
15 | 9, 12, 4 | atcvr0 38986 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
16 | 15 | 3adant2 1128 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
17 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
18 | 3, 17, 12 | cvrcmp 38981 | . 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 1385 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 Basecbs 17213 lecple 17273 Posetcpo 18332 0.cp0 18448 ⋖ ccvr 38960 Atomscatm 38961 AtLatcal 38962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-proset 18320 df-poset 18338 df-plt 18355 df-glb 18372 df-p0 18450 df-lat 18457 df-covers 38964 df-ats 38965 df-atl 38996 |
This theorem is referenced by: atncmp 39010 atnlt 39011 atnle 39015 cvlsupr2 39041 cvratlem 39120 2atjm 39144 atbtwn 39145 2atm 39226 2llnmeqat 39270 dalem25 39397 dalem55 39426 dalem57 39428 snatpsubN 39449 pmapat 39462 2llnma1b 39485 cdlemblem 39492 lhp2at0nle 39734 lhpat3 39745 4atexlemcnd 39771 trlval3 39886 cdlemc5 39894 cdleme3 39936 cdleme7 39948 cdleme11k 39967 cdleme16b 39978 cdleme16e 39981 cdleme16f 39982 cdlemednpq 39998 cdleme20j 40017 cdleme22aa 40038 cdleme22cN 40041 cdleme22d 40042 cdlemf2 40261 cdlemb3 40305 cdlemg12e 40346 cdlemg17dALTN 40363 cdlemg19a 40382 cdlemg27b 40395 cdlemg31d 40399 trlcone 40427 cdlemi 40519 tendotr 40529 cdlemk17 40557 cdlemk52 40653 cdleml1N 40675 dia2dimlem1 40763 dia2dimlem2 40764 dia2dimlem3 40765 |
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