Theorem cdlemg1cN 41329

Description: Any translation belongs to the set of functions constructed for cdleme 41302. 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 1229	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpll2 1230	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpr 489	. . . 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 41327	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)))
9	1, 2, 3, 8	syl3anc 1396	. . 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 780	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑄) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) = 𝑄)
11	10	eqeq2d 2772	. . . 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 7369	. . 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 2796	. 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 41328	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 ∈ 𝑇)
15	14	adantlr 727	. 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 812	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   class class class wbr 5108  ‘cfv 6536  ℩crio 7366  lecple 17316  Atomscatm 40005  HLchlt 40092  LHypclh 40726  LTrncltrn 40843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-riotaBAD 39695

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-undef 8268  df-map 8825  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-p1 18479  df-lat 18487  df-clat 18554  df-oposet 39918  df-ol 39920  df-oml 39921  df-covers 40008  df-ats 40009  df-atl 40040  df-cvlat 40064  df-hlat 40093  df-llines 40240  df-lplanes 40241  df-lvols 40242  df-lines 40243  df-psubsp 40245  df-pmap 40246  df-padd 40538  df-lhyp 40730  df-laut 40731  df-ldil 40846  df-ltrn 40847  df-trl 40901

This theorem is referenced by:  cdlemg2cN  41331