![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemfnid | Structured version Visualization version GIF version |
Description: cdlemf 40545 with additional constraint of non-identity. (Contributed by NM, 20-Jun-2013.) |
Ref | Expression |
---|---|
cdlemfnid.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemfnid.l | ⊢ ≤ = (le‘𝐾) |
cdlemfnid.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemfnid.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemfnid.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemfnid.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemfnid | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemfnid.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdlemfnid.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | cdlemfnid.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | cdlemfnid.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | cdlemfnid.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | cdlemf 40545 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈) |
7 | simp3 1137 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑅‘𝑓) = 𝑈) | |
8 | simp1rl 1237 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑈 ∈ 𝐴) | |
9 | 7, 8 | eqeltrd 2838 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑅‘𝑓) ∈ 𝐴) |
10 | simp1l 1196 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | simp2 1136 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑓 ∈ 𝑇) | |
12 | cdlemfnid.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾) | |
13 | 12, 2, 3, 4, 5 | trlnidatb 40159 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝑓) ∈ 𝐴)) |
14 | 10, 11, 13 | syl2anc 584 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝑓) ∈ 𝐴)) |
15 | 9, 14 | mpbird 257 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑓 ≠ ( I ↾ 𝐵)) |
16 | 7, 15 | jca 511 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵))) |
17 | 16 | 3expia 1120 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑅‘𝑓) = 𝑈 → ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵)))) |
18 | 17 | reximdva 3165 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈 → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵)))) |
19 | 6, 18 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 class class class wbr 5147 I cid 5581 ↾ cres 5690 ‘cfv 6562 Basecbs 17244 lecple 17304 Atomscatm 39244 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 trLctrl 40140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-undef 8296 df-map 8866 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 |
This theorem is referenced by: cdlemftr3 40547 cdlemj3 40805 |
Copyright terms: Public domain | W3C validator |