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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 1187 | . . . 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 37920 | . . . 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 1222 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → 𝑈 ∈ 𝐸) | |
10 | 2, 3, 4, 5, 6 | tendo0mul 37956 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → ( 0 ∘ 𝑈) = 0 ) |
11 | 1, 9, 10 | syl2anc 586 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ( 0 ∘ 𝑈) = 0 ) |
12 | simpr 487 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → 𝑉 = 0 ) | |
13 | 11, 12 | eqtr4d 2859 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ( 0 ∘ 𝑈) = 𝑉) |
14 | coeq1 5722 | . . . . 5 ⊢ (𝑠 = 0 → (𝑠 ∘ 𝑈) = ( 0 ∘ 𝑈)) | |
15 | 14 | eqeq1d 2823 | . . . 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 586 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 = 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
18 | simpl1 1187 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
19 | simpl2 1188 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) | |
20 | simpl3 1189 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → 𝑈 ≠ 0 ) | |
21 | simpr 487 | . . 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 38109 | . . 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 1370 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑈 ≠ 0 ) ∧ 𝑉 ≠ 0 ) → ∃𝑠 ∈ 𝐸 (𝑠 ∘ 𝑈) = 𝑉) |
25 | 17, 24 | pm2.61dane 3104 | 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ↦ cmpt 5138 I cid 5453 ↾ cres 5551 ∘ ccom 5553 ‘cfv 6349 Basecbs 16477 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 TEndoctendo 37882 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-riotaBAD 36083 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-undef 7933 df-map 8402 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tendo 37885 |
This theorem is referenced by: (None) |
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