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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml2N | 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) |
Ref | Expression |
---|---|
cdleml2N | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → ∃𝑠 ∈ 𝐸 (𝑠‘(𝑈‘𝑓)) = (𝑉‘𝑓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1134 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp21 1204 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → 𝑈 ∈ 𝐸) | |
3 | simp23 1206 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → 𝑓 ∈ 𝑇) | |
4 | cdleml1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | cdleml1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | cdleml1.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
7 | 4, 5, 6 | tendocl 38760 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) → (𝑈‘𝑓) ∈ 𝑇) |
8 | 1, 2, 3, 7 | syl3anc 1369 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → (𝑈‘𝑓) ∈ 𝑇) |
9 | simp22 1205 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → 𝑉 ∈ 𝐸) | |
10 | 4, 5, 6 | tendocl 38760 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) → (𝑉‘𝑓) ∈ 𝑇) |
11 | 1, 9, 3, 10 | syl3anc 1369 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → (𝑉‘𝑓) ∈ 𝑇) |
12 | cdleml1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
13 | cdleml1.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
14 | 12, 4, 5, 13, 6 | cdleml1N 38969 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → (𝑅‘(𝑈‘𝑓)) = (𝑅‘(𝑉‘𝑓))) |
15 | 4, 5, 13, 6 | cdlemk 38967 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑈‘𝑓) ∈ 𝑇 ∧ (𝑉‘𝑓) ∈ 𝑇) ∧ (𝑅‘(𝑈‘𝑓)) = (𝑅‘(𝑉‘𝑓))) → ∃𝑠 ∈ 𝐸 (𝑠‘(𝑈‘𝑓)) = (𝑉‘𝑓)) |
16 | 1, 8, 11, 14, 15 | syl121anc 1373 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈‘𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉‘𝑓) ≠ ( I ↾ 𝐵))) → ∃𝑠 ∈ 𝐸 (𝑠‘(𝑈‘𝑓)) = (𝑉‘𝑓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∃wrex 3066 I cid 5487 ↾ cres 5590 ‘cfv 6430 Basecbs 16893 HLchlt 37343 LHypclh 37977 LTrncltrn 38094 trLctrl 38151 TEndoctendo 38745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-riotaBAD 36946 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-undef 8073 df-map 8591 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tendo 38748 |
This theorem is referenced by: cdleml3N 38971 |
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