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Mirrors > Home > MPE Home > Th. List > Mathboxes > osumcllem2N | Structured version Visualization version GIF version |
Description: Lemma for osumclN 37718. (Contributed by NM, 25-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 |
---|---|
osumcllem2N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1193 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝐾 ∈ HL) | |
2 | simpl2 1194 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ 𝐴) | |
3 | simpr 488 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈) | |
4 | 3 | snssd 4722 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → {𝑝} ⊆ 𝑈) |
5 | osumcllem.u | . . . . . 6 ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) | |
6 | osumcllem.a | . . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | osumcllem.p | . . . . . . . . . 10 ⊢ + = (+𝑃‘𝐾) | |
8 | 6, 7 | paddssat 37565 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
9 | 8 | adantr 484 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑋 + 𝑌) ⊆ 𝐴) |
10 | osumcllem.o | . . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
11 | 6, 10 | polssatN 37659 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) |
12 | 1, 9, 11 | syl2anc 587 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) |
13 | 6, 10 | polssatN 37659 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴) |
14 | 1, 12, 13 | syl2anc 587 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴) |
15 | 5, 14 | eqsstrid 3949 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑈 ⊆ 𝐴) |
16 | 4, 15 | sstrd 3911 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → {𝑝} ⊆ 𝐴) |
17 | 6, 7 | sspadd1 37566 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ {𝑝} ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + {𝑝})) |
18 | 1, 2, 16, 17 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑋 + {𝑝})) |
19 | osumcllem.m | . . 3 ⊢ 𝑀 = (𝑋 + {𝑝}) | |
20 | 18, 19 | sseqtrrdi 3952 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ 𝑀) |
21 | osumcllem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
22 | osumcllem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
23 | osumcllem.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
24 | 21, 22, 6, 7, 10, 23, 19, 5 | osumcllem1N 37707 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑈 ∩ 𝑀) = 𝑀) |
25 | 20, 24 | sseqtrrd 3942 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 {csn 4541 ‘cfv 6380 (class class class)co 7213 lecple 16809 joincjn 17818 Atomscatm 37014 HLchlt 37101 +𝑃cpadd 37546 ⊥𝑃cpolN 37653 PSubClcpscN 37685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-riotaBAD 36704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-undef 8015 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-polarityN 37654 |
This theorem is referenced by: osumcllem9N 37715 |
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