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Mirrors > Home > MPE Home > Th. List > Mathboxes > padd4N | Structured version Visualization version GIF version |
Description: Rearrangement of 4 terms in a 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 |
---|---|
padd4N | ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1136 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → 𝐾 ∈ HL) | |
2 | simp2r 1200 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → 𝑌 ⊆ 𝐴) | |
3 | simp3l 1201 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → 𝑍 ⊆ 𝐴) | |
4 | simp3r 1202 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → 𝑊 ⊆ 𝐴) | |
5 | paddass.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | paddass.p | . . . . 5 ⊢ + = (+𝑃‘𝐾) | |
7 | 5, 6 | padd12N 38269 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊))) |
8 | 1, 2, 3, 4, 7 | syl13anc 1372 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊))) |
9 | 8 | oveq2d 7369 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
10 | simp2l 1199 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → 𝑋 ⊆ 𝐴) | |
11 | 5, 6 | paddssat 38244 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴) → (𝑍 + 𝑊) ⊆ 𝐴) |
12 | 1, 3, 4, 11 | syl3anc 1371 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → (𝑍 + 𝑊) ⊆ 𝐴) |
13 | 5, 6 | paddass 38268 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ (𝑍 + 𝑊) ⊆ 𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
14 | 1, 10, 2, 12, 13 | syl13anc 1372 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
15 | 5, 6 | paddssat 38244 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴) → (𝑌 + 𝑊) ⊆ 𝐴) |
16 | 1, 2, 4, 15 | syl3anc 1371 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → (𝑌 + 𝑊) ⊆ 𝐴) |
17 | 5, 6 | paddass 38268 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ∧ (𝑌 + 𝑊) ⊆ 𝐴)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
18 | 1, 10, 3, 16, 17 | syl13anc 1372 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
19 | 9, 14, 18 | 3eqtr4d 2786 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ‘cfv 6493 (class class class)co 7353 Atomscatm 37692 HLchlt 37779 +𝑃cpadd 38225 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-lat 18313 df-clat 18380 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-padd 38226 |
This theorem is referenced by: paddclN 38272 |
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