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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cvratlt | Structured version Visualization version GIF version |
Description: An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.) |
Ref | Expression |
---|---|
1cvratlt.b | ⊢ 𝐵 = (Base‘𝐾) |
1cvratlt.l | ⊢ ≤ = (le‘𝐾) |
1cvratlt.s | ⊢ < = (lt‘𝐾) |
1cvratlt.u | ⊢ 1 = (1.‘𝐾) |
1cvratlt.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
1cvratlt.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
1cvratlt | ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝑃 < 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝐾 ∈ HL) | |
2 | simpl3 1190 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝑋 ∈ 𝐵) | |
3 | simprl 769 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝑋𝐶 1 ) | |
4 | 1cvratlt.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 1cvratlt.s | . . . 4 ⊢ < = (lt‘𝐾) | |
6 | 1cvratlt.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
7 | 1cvratlt.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
8 | 1cvratlt.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 4, 5, 6, 7, 8 | 1cvratex 39172 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋𝐶 1 ) → ∃𝑞 ∈ 𝐴 𝑞 < 𝑋) |
10 | 1, 2, 3, 9 | syl3anc 1368 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 𝑞 < 𝑋) |
11 | simp1l1 1263 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝐾 ∈ HL) | |
12 | simp1l2 1264 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑃 ∈ 𝐴) | |
13 | simp2 1134 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑞 ∈ 𝐴) | |
14 | simp1l3 1265 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑋 ∈ 𝐵) | |
15 | simp1rr 1236 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑃 ≤ 𝑋) | |
16 | simp3 1135 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑞 < 𝑋) | |
17 | 1cvratlt.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
18 | 4, 17, 5, 8 | atlelt 39137 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑞 < 𝑋)) → 𝑃 < 𝑋) |
19 | 11, 12, 13, 14, 15, 16, 18 | syl132anc 1385 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 < 𝑋) → 𝑃 < 𝑋) |
20 | 19 | rexlimdv3a 3149 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → (∃𝑞 ∈ 𝐴 𝑞 < 𝑋 → 𝑃 < 𝑋)) |
21 | 10, 20 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ 𝑃 ≤ 𝑋)) → 𝑃 < 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 class class class wbr 5153 ‘cfv 6554 Basecbs 17213 lecple 17273 ltcplt 18333 1.cp1 18449 ⋖ ccvr 38960 Atomscatm 38961 HLchlt 39048 |
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-ov 7427 df-oprab 7428 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 |
This theorem is referenced by: cdlemb 39493 lhplt 39699 |
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