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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuv2-3N | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma2 (p) is 𝑌, f1 is 𝐷, and k1 is 𝑂. (Contributed by NM, 6-Jul-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemk3.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk3.l | ⊢ ≤ = (le‘𝐾) |
cdlemk3.j | ⊢ ∨ = (join‘𝐾) |
cdlemk3.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk3.s | ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) |
cdlemk3.u1 | ⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑)))))) |
Ref | Expression |
---|---|
cdlemkuv2-3N | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐺)‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp22 1206 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐷 ∈ 𝑇) | |
2 | simp13 1204 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐺 ∈ 𝑇) | |
3 | cdlemk3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
4 | cdlemk3.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
5 | cdlemk3.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | cdlemk3.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
7 | cdlemk3.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | cdlemk3.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | cdlemk3.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | cdlemk3.r | . . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | cdlemk3.s | . . . . 5 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) | |
12 | cdlemk3.u1 | . . . . 5 ⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑)))))) | |
13 | eqid 2738 | . . . . 5 ⊢ (𝑆‘𝐷) = (𝑆‘𝐷) | |
14 | eqid 2738 | . . . . 5 ⊢ (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))) = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))) | |
15 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | cdlemkuu 38909 | . . . 4 ⊢ ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷𝑌𝐺) = ((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))‘𝐺)) |
16 | 1, 2, 15 | syl2anc 584 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐷𝑌𝐺) = ((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))‘𝐺)) |
17 | 16 | fveq1d 6776 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐺)‘𝑃) = (((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))‘𝐺)‘𝑃)) |
18 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14 | cdlemkuv2 38881 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))‘𝐺)‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) |
19 | 17, 18 | eqtrd 2778 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐺)‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ↦ cmpt 5157 I cid 5488 ◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 ∈ cmpo 7277 Basecbs 16912 lecple 16969 joincjn 18029 meetcmee 18030 Atomscatm 37277 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 trLctrl 38172 |
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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967 |
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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 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 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-undef 8089 df-map 8617 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 |
This theorem is referenced by: (None) |
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