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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg1idlemN | Structured version Visualization version GIF version |
Description: Lemma for cdlemg1idN 38599. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemg1.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemg1.l | ⊢ ≤ = (le‘𝐾) |
cdlemg1.j | ⊢ ∨ = (join‘𝐾) |
cdlemg1.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg1.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdlemg1.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdlemg1.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemg1.g | ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) |
cdlemg1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg1.f | ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
Ref | Expression |
---|---|
cdlemg1idlemN | ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemg1.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg1.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg1.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg1.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg1.u | . . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | cdlemg1.d | . . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdlemg1.e | . . . . 5 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemg1.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
11 | cdlemg1.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
12 | cdlemg1.f | . . . . 5 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemg1b2 38593 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 = 𝐺) |
14 | 13 | fveq1d 6768 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
15 | 10 | cdleme31id 38416 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐺‘𝑋) = 𝑋) |
16 | 14, 15 | sylan9eq 2798 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄)) → (𝐹‘𝑋) = 𝑋) |
17 | 16 | anassrs 468 | 1 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⦋csb 3831 ifcif 4459 class class class wbr 5073 ↦ cmpt 5156 ‘cfv 6426 ℩crio 7223 (class class class)co 7267 Basecbs 16922 lecple 16979 joincjn 18039 meetcmee 18040 Atomscatm 37285 HLchlt 37372 LHypclh 38006 LTrncltrn 38123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-riotaBAD 36975 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-undef 8076 df-map 8604 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-llines 37520 df-lplanes 37521 df-lvols 37522 df-lines 37523 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 df-laut 38011 df-ldil 38126 df-ltrn 38127 df-trl 38181 |
This theorem is referenced by: cdlemg1idN 38599 |
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