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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddasslem11 | Structured version Visualization version GIF version |
Description: Lemma for paddass 34142. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.) |
Ref | Expression |
---|---|
paddasslem.l | ⊢ ≤ = (le‘𝐾) |
paddasslem.j | ⊢ ∨ = (join‘𝐾) |
paddasslem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddasslem.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddasslem11 | ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 794 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝐾 ∈ HL) | |
2 | simplr3 1098 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑍 ⊆ 𝐴) | |
3 | simplr1 1096 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑋 ⊆ 𝐴) | |
4 | simplr2 1097 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑌 ⊆ 𝐴) | |
5 | paddasslem.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | paddasslem.p | . . . . 5 ⊢ + = (+𝑃‘𝐾) | |
7 | 5, 6 | paddssat 34118 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
8 | 1, 3, 4, 7 | syl3anc 1318 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → (𝑋 + 𝑌) ⊆ 𝐴) |
9 | 5, 6 | sspadd2 34120 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴 ∧ (𝑋 + 𝑌) ⊆ 𝐴) → 𝑍 ⊆ ((𝑋 + 𝑌) + 𝑍)) |
10 | 1, 2, 8, 9 | syl3anc 1318 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑍 ⊆ ((𝑋 + 𝑌) + 𝑍)) |
11 | simpllr 795 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 = 𝑧) | |
12 | simpr 476 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ 𝑍) | |
13 | 11, 12 | eqeltrd 2688 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ 𝑍) |
14 | 10, 13 | sseldd 3569 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 Atomscatm 33568 HLchlt 33655 +𝑃cpadd 34099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-padd 34100 |
This theorem is referenced by: paddasslem14 34137 |
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