Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkid | Structured version Visualization version GIF version |
Description: The value of the tau function (in Lemma K of [Crawley] p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk5.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk5.l | ⊢ ≤ = (le‘𝐾) |
cdlemk5.j | ⊢ ∨ = (join‘𝐾) |
cdlemk5.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk5.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk5.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk5.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk5.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk5.z | ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) |
cdlemk5.y | ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) |
cdlemk5.x | ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) |
Ref | Expression |
---|---|
cdlemkid | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk5.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
2 | 1 | fvexi 6683 | . 2 ⊢ 𝑇 ∈ V |
3 | nfv 1911 | . . 3 ⊢ Ⅎ𝑏((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) | |
4 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑏𝐺 | |
5 | cdlemk5.x | . . . . . . 7 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) | |
6 | nfra1 3219 | . . . . . . . 8 ⊢ Ⅎ𝑏∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌) | |
7 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑏𝑇 | |
8 | 6, 7 | nfriota 7125 | . . . . . . 7 ⊢ Ⅎ𝑏(℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) |
9 | 5, 8 | nfcxfr 2975 | . . . . . 6 ⊢ Ⅎ𝑏𝑋 |
10 | 4, 9 | nfcsbw 3908 | . . . . 5 ⊢ Ⅎ𝑏⦋𝐺 / 𝑔⦌𝑋 |
11 | 10 | nfeq1 2993 | . . . 4 ⊢ Ⅎ𝑏⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵) |
12 | 11 | a1i 11 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → Ⅎ𝑏⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵)) |
13 | cdlemk5.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
14 | cdlemk5.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
15 | cdlemk5.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
16 | cdlemk5.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
17 | cdlemk5.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
18 | cdlemk5.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
19 | cdlemk5.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
20 | cdlemk5.z | . . . 4 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) | |
21 | cdlemk5.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
22 | 13, 14, 15, 16, 17, 18, 1, 19, 20, 21, 5 | cdlemkid4 38069 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ⦋𝐺 / 𝑔⦌𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)) → 𝑧 = ( I ↾ 𝐵)))) |
23 | eqeq1 2825 | . . . 4 ⊢ (( I ↾ 𝐵) = ⦋𝐺 / 𝑔⦌𝑋 → (( I ↾ 𝐵) = ( I ↾ 𝐵) ↔ ⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵))) | |
24 | 23 | adantl 484 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) ∧ ( I ↾ 𝐵) = ⦋𝐺 / 𝑔⦌𝑋) → (( I ↾ 𝐵) = ( I ↾ 𝐵) ↔ ⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵))) |
25 | eqidd 2822 | . . . 4 ⊢ ((𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺))) → ( I ↾ 𝐵) = ( I ↾ 𝐵)) | |
26 | 25 | a1i 11 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ((𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺))) → ( I ↾ 𝐵) = ( I ↾ 𝐵))) |
27 | 13, 14, 15, 16, 17, 18, 1, 19, 20, 21, 5 | cdlemkid5 38070 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ⦋𝐺 / 𝑔⦌𝑋 ∈ 𝑇) |
28 | 13, 18, 1, 19 | cdlemftr2 37701 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑏 ∈ 𝑇 (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺))) |
29 | 28 | 3ad2ant1 1129 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ∃𝑏 ∈ 𝑇 (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺))) |
30 | 3, 12, 22, 24, 26, 27, 29 | riotasv3d 36095 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) ∧ 𝑇 ∈ V) → ⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵)) |
31 | 2, 30 | mpan2 689 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → ⦋𝐺 / 𝑔⦌𝑋 = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⦋csb 3882 class class class wbr 5065 I cid 5458 ◡ccnv 5553 ↾ cres 5556 ∘ ccom 5558 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 Basecbs 16482 lecple 16571 joincjn 17553 meetcmee 17554 Atomscatm 36398 HLchlt 36485 LHypclh 37119 LTrncltrn 37236 trLctrl 37293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-undef 7938 df-map 8407 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 |
This theorem is referenced by: cdlemk35s-id 38073 cdlemk39s-id 38075 cdlemk53b 38091 cdlemk53 38092 |
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