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Mirrors > Home > MPE Home > Th. List > Mathboxes > pexmidlem2N | Structured version Visualization version GIF version |
Description: Lemma for pexmidN 37099. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pexmidlem.l | ⊢ ≤ = (le‘𝐾) |
pexmidlem.j | ⊢ ∨ = (join‘𝐾) |
pexmidlem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pexmidlem.p | ⊢ + = (+𝑃‘𝐾) |
pexmidlem.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
pexmidlem.m | ⊢ 𝑀 = (𝑋 + {𝑝}) |
Ref | Expression |
---|---|
pexmidlem2N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1187 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝐾 ∈ HL) | |
2 | 1 | hllatd 36494 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝐾 ∈ Lat) |
3 | simpl2 1188 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑋 ⊆ 𝐴) | |
4 | pexmidlem.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | pexmidlem.o | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 4, 5 | polssatN 37038 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
7 | 1, 3, 6 | syl2anc 586 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → ( ⊥ ‘𝑋) ⊆ 𝐴) |
8 | simpr1 1190 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑟 ∈ 𝑋) | |
9 | simpr2 1191 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑞 ∈ ( ⊥ ‘𝑋)) | |
10 | simpl3 1189 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ 𝐴) | |
11 | simpr3 1192 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ≤ (𝑟 ∨ 𝑞)) | |
12 | pexmidlem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | pexmidlem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
14 | pexmidlem.p | . . 3 ⊢ + = (+𝑃‘𝐾) | |
15 | 12, 13, 4, 14 | elpaddri 36932 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) |
16 | 2, 3, 7, 8, 9, 10, 11, 15 | syl322anc 1394 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 lecple 16566 joincjn 17548 Latclat 17649 Atomscatm 36393 HLchlt 36480 +𝑃cpadd 36925 ⊥𝑃cpolN 37032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-undef 7933 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-polarityN 37033 |
This theorem is referenced by: pexmidlem3N 37102 |
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