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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddidm | Structured version Visualization version GIF version |
Description: Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.) |
Ref | Expression |
---|---|
paddidm.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
paddidm.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddidm | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ 𝐵) | |
2 | eqid 2732 | . . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | paddidm.s | . . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾) | |
4 | 2, 3 | psubssat 38613 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
5 | eqid 2732 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | eqid 2732 | . . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | paddidm.p | . . . . . 6 ⊢ + = (+𝑃‘𝐾) | |
8 | 5, 6, 2, 7 | elpadd 38658 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))) |
9 | 1, 4, 4, 8 | syl3anc 1371 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑝 ∈ (𝑋 + 𝑋) ↔ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))) |
10 | pm1.2 902 | . . . . . 6 ⊢ ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝑋) | |
11 | 10 | a1i 11 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → ((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝑋)) |
12 | 5, 6, 2, 3 | psubspi 38606 | . . . . . . 7 ⊢ (((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ 𝑋) |
13 | 12 | 3exp1 1352 | . . . . . 6 ⊢ (𝐾 ∈ 𝐵 → (𝑋 ∈ 𝑆 → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ 𝑋)))) |
14 | 13 | imp4b 422 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ 𝑋)) |
15 | 11, 14 | jaod 857 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (((𝑝 ∈ 𝑋 ∨ 𝑝 ∈ 𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) → 𝑝 ∈ 𝑋)) |
16 | 9, 15 | sylbid 239 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑝 ∈ (𝑋 + 𝑋) → 𝑝 ∈ 𝑋)) |
17 | 16 | ssrdv 3987 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) ⊆ 𝑋) |
18 | 2, 7 | sspadd1 38674 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑋)) |
19 | 1, 4, 4, 18 | syl3anc 1371 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (𝑋 + 𝑋)) |
20 | 17, 19 | eqssd 3998 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 lecple 17200 joincjn 18260 Atomscatm 38121 PSubSpcpsubsp 38355 +𝑃cpadd 38654 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-psubsp 38362 df-padd 38655 |
This theorem is referenced by: paddclN 38701 paddss 38704 pmod1i 38707 |
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