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Mirrors > Home > MPE Home > Th. List > Mathboxes > osumcllem4N | Structured version Visualization version GIF version |
Description: Lemma for osumclN 37263. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
osumcllem.l | ⊢ ≤ = (le‘𝐾) |
osumcllem.j | ⊢ ∨ = (join‘𝐾) |
osumcllem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
osumcllem.p | ⊢ + = (+𝑃‘𝐾) |
osumcllem.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
osumcllem.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
osumcllem.m | ⊢ 𝑀 = (𝑋 + {𝑝}) |
osumcllem.u | ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) |
Ref | Expression |
---|---|
osumcllem4N | ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑞 ≠ 𝑟) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4249 | . . 3 ⊢ (𝑟 ∈ (𝑋 ∩ 𝑌) → ¬ (𝑋 ∩ 𝑌) = ∅) | |
2 | incom 4128 | . . . . . . 7 ⊢ (𝑋 ∩ 𝑌) = (𝑌 ∩ 𝑋) | |
3 | sslin 4161 | . . . . . . . 8 ⊢ (𝑋 ⊆ ( ⊥ ‘𝑌) → (𝑌 ∩ 𝑋) ⊆ (𝑌 ∩ ( ⊥ ‘𝑌))) | |
4 | 3 | 3ad2ant3 1132 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑌 ∩ 𝑋) ⊆ (𝑌 ∩ ( ⊥ ‘𝑌))) |
5 | 2, 4 | eqsstrid 3963 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑋 ∩ 𝑌) ⊆ (𝑌 ∩ ( ⊥ ‘𝑌))) |
6 | osumcllem.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | osumcllem.o | . . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
8 | 6, 7 | pnonsingN 37229 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (𝑌 ∩ ( ⊥ ‘𝑌)) = ∅) |
9 | 8 | 3adant3 1129 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑌 ∩ ( ⊥ ‘𝑌)) = ∅) |
10 | 5, 9 | sseqtrd 3955 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑋 ∩ 𝑌) ⊆ ∅) |
11 | ss0b 4305 | . . . . 5 ⊢ ((𝑋 ∩ 𝑌) ⊆ ∅ ↔ (𝑋 ∩ 𝑌) = ∅) | |
12 | 10, 11 | sylib 221 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑋 ∩ 𝑌) = ∅) |
13 | 12 | adantr 484 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → (𝑋 ∩ 𝑌) = ∅) |
14 | 1, 13 | nsyl3 140 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → ¬ 𝑟 ∈ (𝑋 ∩ 𝑌)) |
15 | simprr 772 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑞 ∈ 𝑌) | |
16 | eleq1w 2872 | . . . . . 6 ⊢ (𝑞 = 𝑟 → (𝑞 ∈ 𝑌 ↔ 𝑟 ∈ 𝑌)) | |
17 | 15, 16 | syl5ibcom 248 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → (𝑞 = 𝑟 → 𝑟 ∈ 𝑌)) |
18 | simprl 770 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑟 ∈ 𝑋) | |
19 | 17, 18 | jctild 529 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → (𝑞 = 𝑟 → (𝑟 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌))) |
20 | elin 3897 | . . . 4 ⊢ (𝑟 ∈ (𝑋 ∩ 𝑌) ↔ (𝑟 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌)) | |
21 | 19, 20 | syl6ibr 255 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → (𝑞 = 𝑟 → 𝑟 ∈ (𝑋 ∩ 𝑌))) |
22 | 21 | necon3bd 3001 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → (¬ 𝑟 ∈ (𝑋 ∩ 𝑌) → 𝑞 ≠ 𝑟)) |
23 | 14, 22 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑞 ≠ 𝑟) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ‘cfv 6324 (class class class)co 7135 lecple 16564 joincjn 17546 Atomscatm 36559 HLchlt 36646 +𝑃cpadd 37091 ⊥𝑃cpolN 37198 PSubClcpscN 37230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-undef 7922 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-pmap 36800 df-polarityN 37199 |
This theorem is referenced by: osumcllem6N 37257 |
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