Theorem cdlemg1cN 39453

Description: Any translation belongs to the set of functions constructed for cdleme 39426. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg1c.l	⊢ ≤ = (le‘𝐾)
cdlemg1c.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg1c.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg1c.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg1cN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) → (𝐹 ∈ 𝑇 ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐹   𝑓,𝐻   𝑓,𝐾   ≤ ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,𝑊

Proof of Theorem cdlemg1cN

Step	Hyp	Ref	Expression
1		simpll1 1212	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpll2 1213	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpr 485	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
4		cdlemg1c.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		cdlemg1c.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg1c.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg1c.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	4, 5, 6, 7	cdlemeiota 39451	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)))
9	1, 2, 3, 8	syl3anc 1371	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)))
10		simplr 767	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) = 𝑄)
11	10	eqeq2d 2743	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → ((𝑓‘𝑃) = (𝐹‘𝑃) ↔ (𝑓‘𝑃) = 𝑄))
12	11	riotabidv 7366	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))
13	9, 12	eqtrd 2772	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))
14	4, 5, 6, 7	cdlemg1ci2 39452	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 ∈ 𝑇)
15	14	adantlr 713	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 ∈ 𝑇)
16	13, 15	impbida 799	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) → (𝐹 ∈ 𝑇 ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6543  ℩crio 7363  lecple 17203  Atomscatm 38128  HLchlt 38215  LHypclh 38850  LTrncltrn 38967

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-riotaBAD 37818

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-undef 8257  df-map 8821  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025

This theorem is referenced by:  cdlemg2cN  39455