Theorem ltrniotavalbN 41420

Description: Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem ltrniotavalbN

Step	Hyp	Ref	Expression
1		simpl1 1210	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1212	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → 𝐹 ∈ 𝑇)
3		simpl2l 1245	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simpl2r 1246	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
5		ltrniotavalb.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		ltrniotavalb.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		ltrniotavalb.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		ltrniotavalb.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		eqid 2765	. . . . 5 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
10	5, 6, 7, 8, 9	ltrniotacl 41415	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) ∈ 𝑇)
11	1, 3, 4, 10	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) ∈ 𝑇)
12		simpr 490	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (𝐹‘𝑃) = 𝑄)
13	5, 6, 7, 8, 9	ltrniotaval 41417	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃) = 𝑄)
14	1, 3, 4, 13	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃) = 𝑄)
15	12, 14	eqtr4d 2803	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → (𝐹‘𝑃) = ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃))
16	5, 6, 7, 8	cdlemd 41043	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹‘𝑃) = ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃)) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))
17	1, 2, 11, 3, 15, 16	syl311anc 1411	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ (𝐹‘𝑃) = 𝑄) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))
18		fveq1 6884	. . 3 ⊢ (𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) → (𝐹‘𝑃) = ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃))
19		simp1 1154	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simp2l 1218	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
21		simp2r 1219	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
22	19, 20, 21, 13	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)‘𝑃) = 𝑄)
23	18, 22	sylan9eqr 2822	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → (𝐹‘𝑃) = 𝑄)
24	17, 23	impbida 813	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) = 𝑄 ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ‘cfv 6540  ℩crio 7375  lecple 17343  Atomscatm 40099  HLchlt 40186  LHypclh 40820  LTrncltrn 40937

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-riotaBAD 39789

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-undef 8275  df-map 8832  df-proset 18376  df-poset 18395  df-plt 18410  df-lub 18426  df-glb 18427  df-join 18428  df-meet 18429  df-p0 18505  df-p1 18506  df-lat 18514  df-clat 18581  df-oposet 40012  df-ol 40014  df-oml 40015  df-covers 40102  df-ats 40103  df-atl 40134  df-cvlat 40158  df-hlat 40187  df-llines 40334  df-lplanes 40335  df-lvols 40336  df-lines 40337  df-psubsp 40339  df-pmap 40340  df-padd 40632  df-lhyp 40824  df-laut 40825  df-ldil 40940  df-ltrn 40941  df-trl 40995

This theorem is used by: (None)