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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcocnvat | Structured version Visualization version GIF version |
Description: Commonly used special case of trlcoat 36793. (Contributed by NM, 1-Jul-2013.) |
Ref | Expression |
---|---|
trlcoat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trlcoat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlcoat.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlcoat.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlcocnvat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ ◡𝐺)) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1170 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp2l 1260 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → 𝐹 ∈ 𝑇) | |
3 | simp2r 1261 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → 𝐺 ∈ 𝑇) | |
4 | trlcoat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | trlcoat.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | 4, 5 | ltrncnv 36216 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
7 | 1, 3, 6 | syl2anc 579 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ◡𝐺 ∈ 𝑇) |
8 | simp3 1172 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘𝐹) ≠ (𝑅‘𝐺)) | |
9 | trlcoat.r | . . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | 4, 5, 9 | trlcnv 36235 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘◡𝐺) = (𝑅‘𝐺)) |
11 | 1, 3, 10 | syl2anc 579 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘◡𝐺) = (𝑅‘𝐺)) |
12 | 8, 11 | neeqtrrd 3073 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘𝐹) ≠ (𝑅‘◡𝐺)) |
13 | trlcoat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
14 | 13, 4, 5, 9 | trlcoat 36793 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ◡𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘◡𝐺)) → (𝑅‘(𝐹 ∘ ◡𝐺)) ∈ 𝐴) |
15 | 1, 2, 7, 12, 14 | syl121anc 1498 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ ◡𝐺)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ◡ccnv 5345 ∘ ccom 5350 ‘cfv 6127 Atomscatm 35333 HLchlt 35420 LHypclh 36054 LTrncltrn 36171 trLctrl 36228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-riotaBAD 35023 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-undef 7669 df-map 8129 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-llines 35568 df-lplanes 35569 df-lvols 35570 df-lines 35571 df-psubsp 35573 df-pmap 35574 df-padd 35866 df-lhyp 36058 df-laut 36059 df-ldil 36174 df-ltrn 36175 df-trl 36229 |
This theorem is referenced by: cdlemh1 36885 cdlemk3 36903 cdlemk6 36907 cdlemk7 36918 cdlemk12 36920 cdlemkole 36923 cdlemk14 36924 cdlemk15 36925 cdlemk5u 36931 cdlemk6u 36932 cdlemk7u 36940 cdlemk12u 36942 cdlemkfid1N 36991 |
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