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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 30126 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 36439 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | 3ad2ant1 1129 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2823 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atcmp.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 36427 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1130 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 4 | atbase 36427 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
8 | 7 | 3ad2ant3 1131 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ (Base‘𝐾)) |
9 | eqid 2823 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
10 | 3, 9 | atl0cl 36441 | . . 3 ⊢ (𝐾 ∈ AtLat → (0.‘𝐾) ∈ (Base‘𝐾)) |
11 | 10 | 3ad2ant1 1129 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾) ∈ (Base‘𝐾)) |
12 | eqid 2823 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 9, 12, 4 | atcvr0 36426 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
14 | 13 | 3adant3 1128 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
15 | 9, 12, 4 | atcvr0 36426 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
16 | 15 | 3adant2 1127 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
17 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
18 | 3, 17, 12 | cvrcmp 36421 | . 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 1384 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 Posetcpo 17552 0.cp0 17649 ⋖ ccvr 36400 Atomscatm 36401 AtLatcal 36402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-proset 17540 df-poset 17558 df-plt 17570 df-glb 17587 df-p0 17651 df-lat 17658 df-covers 36404 df-ats 36405 df-atl 36436 |
This theorem is referenced by: atncmp 36450 atnlt 36451 atnle 36455 cvlsupr2 36481 cvratlem 36559 2atjm 36583 atbtwn 36584 2atm 36665 2llnmeqat 36709 dalem25 36836 dalem55 36865 dalem57 36867 snatpsubN 36888 pmapat 36901 2llnma1b 36924 cdlemblem 36931 lhp2at0nle 37173 lhpat3 37184 4atexlemcnd 37210 trlval3 37325 cdlemc5 37333 cdleme3 37375 cdleme7 37387 cdleme11k 37406 cdleme16b 37417 cdleme16e 37420 cdleme16f 37421 cdlemednpq 37437 cdleme20j 37456 cdleme22aa 37477 cdleme22cN 37480 cdleme22d 37481 cdlemf2 37700 cdlemb3 37744 cdlemg12e 37785 cdlemg17dALTN 37802 cdlemg19a 37821 cdlemg27b 37834 cdlemg31d 37838 trlcone 37866 cdlemi 37958 tendotr 37968 cdlemk17 37996 cdlemk52 38092 cdleml1N 38114 dia2dimlem1 38202 dia2dimlem2 38203 dia2dimlem3 38204 |
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