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Mirrors > Home > MPE Home > Th. List > Mathboxes > osumcllem5N | Structured version Visualization version GIF version |
Description: Lemma for osumclN 35571. (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 |
---|---|
osumcllem5N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp11 1111 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝐾 ∈ HL) | |
2 | hllat 34968 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝐾 ∈ Lat) |
4 | simp12 1112 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑋 ⊆ 𝐴) | |
5 | simp13 1113 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑌 ⊆ 𝐴) | |
6 | simp31 1117 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑟 ∈ 𝑋) | |
7 | simp32 1118 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑞 ∈ 𝑌) | |
8 | simp2 1082 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ 𝐴) | |
9 | simp33 1119 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ≤ (𝑟 ∨ 𝑞)) | |
10 | osumcllem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
11 | osumcllem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
12 | osumcllem.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | osumcllem.p | . . 3 ⊢ + = (+𝑃‘𝐾) | |
14 | 10, 11, 12, 13 | elpaddri 35406 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌)) |
15 | 3, 4, 5, 6, 7, 8, 9, 14 | syl322anc 1394 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 {csn 4210 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 lecple 15995 joincjn 16991 Latclat 17092 Atomscatm 34868 HLchlt 34955 +𝑃cpadd 35399 ⊥𝑃cpolN 35506 PSubClcpscN 35538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-lub 17021 df-join 17023 df-lat 17093 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-padd 35400 |
This theorem is referenced by: osumcllem6N 35565 |
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