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Theorem cdlemg1cex 39051
Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 39026? (Contributed by NM, 17-Apr-2013.)
Hypotheses
Ref Expression
cdlemg1c.l ≀ = (leβ€˜πΎ)
cdlemg1c.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg1c.h 𝐻 = (LHypβ€˜πΎ)
cdlemg1c.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemg1cex ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
Distinct variable groups:   𝑓,𝑝,π‘ž,𝐴   𝑓,𝐹,𝑝,π‘ž   𝑓,𝐻,𝑝,π‘ž   𝑓,𝐾,𝑝,π‘ž   ≀ ,𝑓,𝑝,π‘ž   𝑇,𝑓,𝑝,π‘ž   𝑓,π‘Š,𝑝,π‘ž

Proof of Theorem cdlemg1cex
StepHypRef Expression
1 cdlemg1c.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
2 cdlemg1c.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
3 cdlemg1c.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
4 cdlemg1c.t . . . . . . . 8 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4ltrnel 38602 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ ((πΉβ€˜π‘) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
653expa 1118 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ ((πΉβ€˜π‘) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
76simpld 495 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (πΉβ€˜π‘) ∈ 𝐴)
8 simprr 771 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ Β¬ 𝑝 ≀ π‘Š)
96simprd 496 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ Β¬ (πΉβ€˜π‘) ≀ π‘Š)
10 simpll 765 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
11 simpr 485 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))
12 simplr 767 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ 𝐹 ∈ 𝑇)
131, 2, 3, 4cdlemeiota 39048 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1410, 11, 12, 13syl3anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
15 breq1 5108 . . . . . . . 8 (π‘ž = (πΉβ€˜π‘) β†’ (π‘ž ≀ π‘Š ↔ (πΉβ€˜π‘) ≀ π‘Š))
1615notbid 317 . . . . . . 7 (π‘ž = (πΉβ€˜π‘) β†’ (Β¬ π‘ž ≀ π‘Š ↔ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
17 eqeq2 2748 . . . . . . . . 9 (π‘ž = (πΉβ€˜π‘) β†’ ((π‘“β€˜π‘) = π‘ž ↔ (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1817riotabidv 7315 . . . . . . . 8 (π‘ž = (πΉβ€˜π‘) β†’ (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž) = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1918eqeq2d 2747 . . . . . . 7 (π‘ž = (πΉβ€˜π‘) β†’ (𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž) ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘))))
2016, 193anbi23d 1439 . . . . . 6 (π‘ž = (πΉβ€˜π‘) β†’ ((Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) ↔ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))))
2120rspcev 3581 . . . . 5 (((πΉβ€˜π‘) ∈ 𝐴 ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
227, 8, 9, 14, 21syl13anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
231, 2, 3lhpexnle 38469 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ≀ π‘Š)
2423adantr 481 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ≀ π‘Š)
2522, 24reximddv 3168 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
2625ex 413 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
27 simp1 1136 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
28 simp2l 1199 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝑝 ∈ 𝐴)
29 simp31 1209 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ Β¬ 𝑝 ≀ π‘Š)
3028, 29jca 512 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))
31 simp2r 1200 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ π‘ž ∈ 𝐴)
32 simp32 1210 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ Β¬ π‘ž ≀ π‘Š)
3331, 32jca 512 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (π‘ž ∈ 𝐴 ∧ Β¬ π‘ž ≀ π‘Š))
34 simp33 1211 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))
351, 2, 3, 4cdlemg1ci2 39049 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š) ∧ (π‘ž ∈ 𝐴 ∧ Β¬ π‘ž ≀ π‘Š)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇)
3627, 30, 33, 34, 35syl31anc 1373 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝐹 ∈ 𝑇)
37363exp 1119 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) β†’ ((Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇)))
3837rexlimdvv 3204 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇))
3926, 38impbid 211 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3073   class class class wbr 5105  β€˜cfv 6496  β„©crio 7312  lecple 17140  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LTrncltrn 38564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-undef 8204  df-map 8767  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622
This theorem is referenced by:  cdlemg2cex  39054
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