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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk35s | Structured version Visualization version GIF version |
Description: Substitution version of cdlemk35 38612. (Contributed by NM, 22-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 |
---|---|
cdlemk35s | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → ⦋𝐺 / 𝑔⦌𝑋 ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp22l 1294 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → 𝐺 ∈ 𝑇) | |
2 | cdlemk5.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cdlemk5.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemk5.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
5 | cdlemk5.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
6 | cdlemk5.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | cdlemk5.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | cdlemk5.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
9 | cdlemk5.r | . . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | cdlemk5.z | . . . . . 6 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) | |
11 | cdlemk5.y | . . . . . 6 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
12 | cdlemk5.x | . . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) | |
13 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemk35 38612 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → 𝑋 ∈ 𝑇) |
14 | 13 | sbcth 3698 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → [𝐺 / 𝑔](((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → 𝑋 ∈ 𝑇)) |
15 | sbcimg 3734 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔](((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → 𝑋 ∈ 𝑇) ↔ ([𝐺 / 𝑔]((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → [𝐺 / 𝑔]𝑋 ∈ 𝑇))) | |
16 | 14, 15 | mpbid 235 | . . 3 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → [𝐺 / 𝑔]𝑋 ∈ 𝑇)) |
17 | eleq1 2818 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝑇 ↔ 𝐺 ∈ 𝑇)) | |
18 | neeq1 2994 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ≠ ( I ↾ 𝐵) ↔ 𝐺 ≠ ( I ↾ 𝐵))) | |
19 | 17, 18 | anbi12d 634 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ↔ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)))) |
20 | 19 | 3anbi2d 1443 | . . . . 5 ⊢ (𝑔 = 𝐺 → (((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ↔ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇))) |
21 | 20 | 3anbi2d 1443 | . . . 4 ⊢ (𝑔 = 𝐺 → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) ↔ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))))) |
22 | 21 | sbcieg 3723 | . . 3 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) ↔ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))))) |
23 | sbcel1g 4314 | . . 3 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝑋 ∈ 𝑇 ↔ ⦋𝐺 / 𝑔⦌𝑋 ∈ 𝑇)) | |
24 | 16, 22, 23 | 3imtr3d 296 | . 2 ⊢ (𝐺 ∈ 𝑇 → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → ⦋𝐺 / 𝑔⦌𝑋 ∈ 𝑇)) |
25 | 1, 24 | mpcom 38 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → ⦋𝐺 / 𝑔⦌𝑋 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 [wsbc 3683 ⦋csb 3798 class class class wbr 5039 I cid 5439 ◡ccnv 5535 ↾ cres 5538 ∘ ccom 5540 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 Basecbs 16666 lecple 16756 joincjn 17772 meetcmee 17773 Atomscatm 36963 HLchlt 37050 LHypclh 37684 LTrncltrn 37801 trLctrl 37858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-riotaBAD 36653 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-undef 7993 df-map 8488 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 |
This theorem is referenced by: cdlemk35s-id 38638 cdlemk47 38649 cdlemk48 38650 cdlemk49 38651 cdlemk50 38652 cdlemk51 38653 cdlemk52 38654 cdlemk53a 38655 |
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