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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0fN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdleme0.l | ⊢ ≤ = (le‘𝐾) |
cdleme0.j | ⊢ ∨ = (join‘𝐾) |
cdleme0.m | ⊢ ∧ = (meet‘𝐾) |
cdleme0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme0c.3 | ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme0fN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≠ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme0c.3 | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊) | |
2 | simp1l 1195 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ HL) | |
3 | 2 | hllatd 38773 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ Lat) |
4 | simp2l 1197 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ 𝐴) | |
5 | eqid 2727 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | cdleme0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atbase 38698 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾)) |
9 | simp3r 1200 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
10 | 5, 6 | atbase 38698 | . . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ (Base‘𝐾)) |
12 | cdleme0.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
13 | 5, 12 | latjcl 18422 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾)) |
14 | 3, 8, 11, 13 | syl3anc 1369 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾)) |
15 | simp1r 1196 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑊 ∈ 𝐻) | |
16 | cdleme0.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
17 | 5, 16 | lhpbase 39408 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
18 | 15, 17 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑊 ∈ (Base‘𝐾)) |
19 | cdleme0.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
20 | cdleme0.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
21 | 5, 19, 20 | latmle2 18448 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊) |
22 | 3, 14, 18, 21 | syl3anc 1369 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊) |
23 | 1, 22 | eqbrtrid 5177 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≤ 𝑊) |
24 | simp2r 1198 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑃 ≤ 𝑊) | |
25 | nbrne2 5162 | . 2 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑉 ≠ 𝑃) | |
26 | 23, 24, 25 | syl2anc 583 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑉 ≠ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 lecple 17231 joincjn 18294 meetcmee 18295 Latclat 18414 Atomscatm 38672 HLchlt 38759 LHypclh 39394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-lat 18415 df-ats 38676 df-atl 38707 df-cvlat 38731 df-hlat 38760 df-lhyp 39398 |
This theorem is referenced by: (None) |
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