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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 28590 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 33606 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | 3ad2ant1 1075 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atcmp.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 33594 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1076 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 4 | atbase 33594 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
8 | 7 | 3ad2ant3 1077 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ (Base‘𝐾)) |
9 | eqid 2610 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
10 | 3, 9 | atl0cl 33608 | . . 3 ⊢ (𝐾 ∈ AtLat → (0.‘𝐾) ∈ (Base‘𝐾)) |
11 | 10 | 3ad2ant1 1075 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾) ∈ (Base‘𝐾)) |
12 | eqid 2610 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 9, 12, 4 | atcvr0 33593 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
14 | 13 | 3adant3 1074 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
15 | 9, 12, 4 | atcvr0 33593 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
16 | 15 | 3adant2 1073 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
17 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
18 | 3, 17, 12 | cvrcmp 33588 | . 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 1336 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 0.cp0 16860 ⋖ ccvr 33567 Atomscatm 33568 AtLatcal 33569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-preset 16751 df-poset 16769 df-plt 16781 df-glb 16798 df-p0 16862 df-lat 16869 df-covers 33571 df-ats 33572 df-atl 33603 |
This theorem is referenced by: atncmp 33617 atnlt 33618 atnle 33622 cvlsupr2 33648 cvratlem 33725 2atjm 33749 atbtwn 33750 2atm 33831 2llnmeqat 33875 dalem25 34002 dalem55 34031 dalem57 34033 snatpsubN 34054 pmapat 34067 2llnma1b 34090 cdlemblem 34097 lhp2at0nle 34339 lhpat3 34350 4atexlemcnd 34376 trlval3 34492 cdlemc5 34500 cdleme3 34542 cdleme7 34554 cdleme11k 34573 cdleme16b 34584 cdleme16e 34587 cdleme16f 34588 cdlemednpq 34604 cdleme20j 34624 cdleme22aa 34645 cdleme22cN 34648 cdleme22d 34649 cdlemf2 34868 cdlemb3 34912 cdlemg12e 34953 cdlemg17dALTN 34970 cdlemg19a 34989 cdlemg27b 35002 cdlemg31d 35006 trlcone 35034 cdlemi 35126 tendotr 35136 cdlemk17 35164 cdlemk52 35260 cdleml1N 35282 dia2dimlem1 35371 dia2dimlem2 35372 dia2dimlem3 35373 |
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