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| Mirrors > Home > MPE Home > Th. List > Mathboxes > padd12N | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| paddass.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| paddass.p | ⊢ + = (+𝑃‘𝐾) | 
| Ref | Expression | 
|---|---|
| padd12N | ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hllat 39365 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
| 2 | 1 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ Lat) | 
| 3 | simpr1 1194 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ 𝐴) | |
| 4 | simpr2 1195 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑌 ⊆ 𝐴) | |
| 5 | paddass.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 6 | paddass.p | . . . . 5 ⊢ + = (+𝑃‘𝐾) | |
| 7 | 5, 6 | paddcom 39816 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | 
| 8 | 2, 3, 4, 7 | syl3anc 1372 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | 
| 9 | 8 | oveq1d 7447 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑌 + 𝑋) + 𝑍)) | 
| 10 | 5, 6 | paddass 39841 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | 
| 11 | simpl 482 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ HL) | |
| 12 | simpr3 1196 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑍 ⊆ 𝐴) | |
| 13 | 5, 6 | paddass 39841 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) | 
| 14 | 11, 4, 3, 12, 13 | syl13anc 1373 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) | 
| 15 | 9, 10, 14 | 3eqtr3d 2784 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 Latclat 18477 Atomscatm 39265 HLchlt 39352 +𝑃cpadd 39798 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-lat 18478 df-clat 18545 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-padd 39799 | 
| This theorem is referenced by: padd4N 39843 pmodl42N 39854 | 
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