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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml5N | Structured version Visualization version GIF version |
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdleml1.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleml1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleml1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdleml1.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdleml1.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
cdleml3.o | ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
cdleml5N | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1189 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | cdleml1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cdleml1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | cdleml1.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | cdleml1.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
6 | cdleml3.o | . . . . 5 ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
7 | 2, 3, 4, 5, 6 | tendo0cl 40734 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸) |
8 | 1, 7 | syl 17 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → 0 ∈ 𝐸) |
9 | simpl2l 1224 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → 𝑈 ∈ 𝐸) | |
10 | 2, 3, 4, 5, 6 | tendo0mul 40770 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → ( 0 ∘ 𝑈) = 0 ) |
11 | 1, 9, 10 | syl2anc 583 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ( 0 ∘ 𝑈) = 0 ) |
12 | simpr 484 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → 𝑉 = 0 ) | |
13 | 11, 12 | eqtr4d 2776 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ( 0 ∘ 𝑈) = 𝑉) |
14 | coeq1 5865 | . . . . 5 ⊢ (𝑠 = 0 → (𝑠 ∘ 𝑈) = ( 0 ∘ 𝑈)) | |
15 | 14 | eqeq1d 2735 | . . . 4 ⊢ (𝑠 = 0 → ((𝑠 ∘ 𝑈) = 𝑉 ↔ ( 0 ∘ 𝑈) = 𝑉)) |
16 | 15 | rspcev 3622 | . . 3 ⊢ (( 0 ∈ 𝐸 ∧ ( 0 ∘ 𝑈) = 𝑉) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
17 | 8, 13, 16 | syl2anc 583 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
18 | simpl1 1189 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
19 | simpl2 1190 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) | |
20 | simpl3 1191 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → 𝑈 ≠ 0 ) | |
21 | simpr 484 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → 𝑉 ≠ 0 ) | |
22 | cdleml1.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
23 | 2, 3, 4, 22, 5, 6 | cdleml4N 40923 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝑈 ≠ 0 ∧ 𝑉 ≠ 0 )) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
24 | 18, 19, 20, 21, 23 | syl112anc 1372 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
25 | 17, 24 | pm2.61dane 3025 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ∃wrex 3066 ↦ cmpt 5232 I cid 5575 ↾ cres 5685 ∘ ccom 5687 ‘cfv 6558 Basecbs 17234 HLchlt 39293 LHypclh 39928 LTrncltrn 40045 trLctrl 40102 TEndoctendo 40696 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-riotaBAD 38896 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-undef 8291 df-map 8861 df-proset 18341 df-poset 18359 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-oposet 39119 df-ol 39121 df-oml 39122 df-covers 39209 df-ats 39210 df-atl 39241 df-cvlat 39265 df-hlat 39294 df-llines 39442 df-lplanes 39443 df-lvols 39444 df-lines 39445 df-psubsp 39447 df-pmap 39448 df-padd 39740 df-lhyp 39932 df-laut 39933 df-ldil 40048 df-ltrn 40049 df-trl 40103 df-tendo 40699 |
This theorem is referenced by: (None) |
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