|   | Mathbox for Norm Megill | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > osumcllem2N | Structured version Visualization version GIF version | ||
| Description: Lemma for osumclN 39969. (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 1192 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝐾 ∈ HL) | |
| 2 | simpl2 1193 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ 𝐴) | |
| 3 | simpr 484 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈) | |
| 4 | 3 | snssd 4809 | . . . . 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 39816 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) | 
| 9 | 8 | adantr 480 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑋 + 𝑌) ⊆ 𝐴) | 
| 10 | osumcllem.o | . . . . . . . . 9 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
| 11 | 6, 10 | polssatN 39910 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) | 
| 12 | 1, 9, 11 | syl2anc 584 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) | 
| 13 | 6, 10 | polssatN 39910 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴) | 
| 14 | 1, 12, 13 | syl2anc 584 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴) | 
| 15 | 5, 14 | eqsstrid 4022 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑈 ⊆ 𝐴) | 
| 16 | 4, 15 | sstrd 3994 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → {𝑝} ⊆ 𝐴) | 
| 17 | 6, 7 | sspadd1 39817 | . . . 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 4025 | . 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 39958 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑈 ∩ 𝑀) = 𝑀) | 
| 25 | 20, 24 | sseqtrrd 4021 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 Atomscatm 39264 HLchlt 39351 +𝑃cpadd 39797 ⊥𝑃cpolN 39904 PSubClcpscN 39936 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-polarityN 39905 | 
| This theorem is referenced by: osumcllem9N 39966 | 
| Copyright terms: Public domain | W3C validator |