| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > atcmp | Structured version Visualization version GIF version | ||
| Description: If two atoms are comparable, they are equal. (atsseq 32366 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 39302 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
| 2 | 1 | 3ad2ant1 1134 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Poset) |
| 3 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 4 | atcmp.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 3, 4 | atbase 39290 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 6 | 5 | 3ad2ant2 1135 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
| 7 | 3, 4 | atbase 39290 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
| 8 | 7 | 3ad2ant3 1136 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ (Base‘𝐾)) |
| 9 | eqid 2737 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
| 10 | 3, 9 | atl0cl 39304 | . . 3 ⊢ (𝐾 ∈ AtLat → (0.‘𝐾) ∈ (Base‘𝐾)) |
| 11 | 10 | 3ad2ant1 1134 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾) ∈ (Base‘𝐾)) |
| 12 | eqid 2737 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 13 | 9, 12, 4 | atcvr0 39289 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
| 14 | 13 | 3adant3 1133 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃) |
| 15 | 9, 12, 4 | atcvr0 39289 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
| 16 | 15 | 3adant2 1132 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑄) |
| 17 | atcmp.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 18 | 3, 17, 12 | cvrcmp 39284 | . 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 1390 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 0.cp0 18468 ⋖ ccvr 39263 Atomscatm 39264 AtLatcal 39265 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-lat 18477 df-covers 39267 df-ats 39268 df-atl 39299 |
| This theorem is referenced by: atncmp 39313 atnlt 39314 atnle 39318 cvlsupr2 39344 cvratlem 39423 2atjm 39447 atbtwn 39448 2atm 39529 2llnmeqat 39573 dalem25 39700 dalem55 39729 dalem57 39731 snatpsubN 39752 pmapat 39765 2llnma1b 39788 cdlemblem 39795 lhp2at0nle 40037 lhpat3 40048 4atexlemcnd 40074 trlval3 40189 cdlemc5 40197 cdleme3 40239 cdleme7 40251 cdleme11k 40270 cdleme16b 40281 cdleme16e 40284 cdleme16f 40285 cdlemednpq 40301 cdleme20j 40320 cdleme22aa 40341 cdleme22cN 40344 cdleme22d 40345 cdlemf2 40564 cdlemb3 40608 cdlemg12e 40649 cdlemg17dALTN 40666 cdlemg19a 40685 cdlemg27b 40698 cdlemg31d 40702 trlcone 40730 cdlemi 40822 tendotr 40832 cdlemk17 40860 cdlemk52 40956 cdleml1N 40978 dia2dimlem1 41066 dia2dimlem2 41067 dia2dimlem3 41068 |
| Copyright terms: Public domain | W3C validator |