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Mirrors > Home > MPE Home > Th. List > Mathboxes > pexmidlem5N | Structured version Visualization version GIF version |
Description: Lemma for pexmidN 37120. (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 |
---|---|
pexmidlem5N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4310 | . . . 4 ⊢ ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ ∅ ↔ ∃𝑞 𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀)) | |
2 | pexmidlem.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
3 | pexmidlem.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
4 | pexmidlem.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | pexmidlem.p | . . . . . . 7 ⊢ + = (+𝑃‘𝐾) | |
6 | pexmidlem.o | . . . . . . 7 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
7 | pexmidlem.m | . . . . . . 7 ⊢ 𝑀 = (𝑋 + {𝑝}) | |
8 | 2, 3, 4, 5, 6, 7 | pexmidlem4N 37124 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) |
9 | 8 | expr 459 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) |
10 | 9 | exlimdv 1934 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (∃𝑞 𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) |
11 | 1, 10 | syl5bi 244 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ ∅ → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) |
12 | 11 | necon1bd 3034 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅)) |
13 | 12 | impr 457 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 ‘cfv 6355 (class class class)co 7156 lecple 16572 joincjn 17554 Atomscatm 36414 HLchlt 36501 +𝑃cpadd 36946 ⊥𝑃cpolN 37053 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
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 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-undef 7939 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-polarityN 37054 |
This theorem is referenced by: pexmidlem6N 37126 |
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