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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemd | Structured version Visualization version GIF version |
Description: If two translations agree at any atom not under the fiducial co-atom 𝑊, then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.) |
Ref | Expression |
---|---|
cdlemd.l | ⊢ ≤ = (le‘𝐾) |
cdlemd.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemd.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemd.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemd | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → 𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl11 1248 | . . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simpl12 1249 | . . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → 𝐹 ∈ 𝑇) | |
3 | simpl13 1250 | . . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → 𝐺 ∈ 𝑇) | |
4 | 2, 3 | jca 511 | . . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) |
5 | simpr 484 | . . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
6 | simpl2 1192 | . . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
7 | simpl3 1193 | . . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → (𝐹‘𝑃) = (𝐺‘𝑃)) | |
8 | cdlemd.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
9 | eqid 2740 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
10 | cdlemd.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | cdlemd.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
12 | cdlemd.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
13 | 8, 9, 10, 11, 12 | cdlemd9 40163 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑞 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐹‘𝑞) = (𝐺‘𝑞)) |
14 | 1, 4, 5, 6, 7, 13 | syl311anc 1384 | . . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) ∧ 𝑞 ∈ 𝐴) → (𝐹‘𝑞) = (𝐺‘𝑞)) |
15 | 14 | ralrimiva 3152 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ∀𝑞 ∈ 𝐴 (𝐹‘𝑞) = (𝐺‘𝑞)) |
16 | 10, 11, 12 | ltrneq2 40105 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑞 ∈ 𝐴 (𝐹‘𝑞) = (𝐺‘𝑞) ↔ 𝐹 = 𝐺)) |
17 | 16 | 3ad2ant1 1133 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (∀𝑞 ∈ 𝐴 (𝐹‘𝑞) = (𝐺‘𝑞) ↔ 𝐹 = 𝐺)) |
18 | 15, 17 | mpbid 232 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 ‘cfv 6573 lecple 17318 joincjn 18381 Atomscatm 39219 HLchlt 39306 LHypclh 39941 LTrncltrn 40058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 |
This theorem is referenced by: ltrneq3 40165 cdleme 40517 cdlemg1a 40527 ltrniotavalbN 40541 cdlemg44 40690 cdlemk19 40826 cdlemk27-3 40864 cdlemk33N 40866 cdlemk34 40867 cdlemk53a 40912 cdlemk19u 40927 dia2dimlem4 41024 dih1dimatlem0 41285 |
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