Theorem cdleme50ltrn 41534

Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐹 is a lattice translation. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)

Hypotheses

Ref	Expression
cdlemef50.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef50.l	⊢ ≤ = (le‘𝐾)
cdlemef50.j	⊢ ∨ = (join‘𝐾)
cdlemef50.m	⊢ ∧ = (meet‘𝐾)
cdlemef50.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef50.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef50.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef50.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs50.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef50.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
cdleme50ltrn.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdleme50ltrn	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧, ∧   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐷(𝑡)   𝑇(𝑥, 𝑦, 𝑧, 𝑡, 𝑠)   𝐸(𝑡, 𝑠)   𝐹(𝑥, 𝑦, 𝑧, 𝑡, 𝑠)

Proof of Theorem cdleme50ltrn

Dummy variables 𝑒 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemef50.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef50.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemef50.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemef50.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemef50.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef50.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef50.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef50.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemefs50.e	. . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemef50.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
11		eqid 2760	. . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cdleme50ldil 41525	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
13		simp1 1154	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
14		simp2l 1218	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → 𝑑 ∈ 𝐴)
15		simp3l 1220	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ¬ 𝑑 ≤ 𝑊)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50trn123 41531	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ ¬ 𝑑 ≤ 𝑊)) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = 𝑈)
17	13, 14, 15, 16	syl12anc 850	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = 𝑈)
18		simp2r 1219	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → 𝑒 ∈ 𝐴)
19		simp3r 1221	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ¬ 𝑒 ≤ 𝑊)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50trn123 41531	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑒 ∈ 𝐴 ∧ ¬ 𝑒 ≤ 𝑊)) → ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊) = 𝑈)
21	13, 18, 19, 20	syl12anc 850	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊) = 𝑈)
22	17, 21	eqtr4d 2798	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ (¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊)) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊))
23	22	3exp 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) → ((¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊))))
24	23	ralrimivv 3203	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ((¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊)))
25		cdleme50ltrn.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
26	2, 3, 4, 5, 6, 11, 25	isltrn 41096	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ((¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊)))))
27	26	3ad2ant1 1151	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ((¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊) → ((𝑑 ∨ (𝐹‘𝑑)) ∧ 𝑊) = ((𝑒 ∨ (𝐹‘𝑒)) ∧ 𝑊)))))
28	12, 24, 27	mpbir2and 726	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ⦋csb 3846  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6527  ℩crio 7364  (class class class)co 7408  Basecbs 17348  lecple 17396  joincjn 18446  meetcmee 18447  Atomscatm 40240  HLchlt 40327  LHypclh 40961  LDilcldil 41077  LTrncltrn 41078

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-riotaBAD 39930

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-undef 8268  df-map 8827  df-proset 18429  df-poset 18448  df-plt 18463  df-lub 18479  df-glb 18480  df-join 18481  df-meet 18482  df-p0 18558  df-p1 18559  df-lat 18567  df-clat 18634  df-oposet 40153  df-ol 40155  df-oml 40156  df-covers 40243  df-ats 40244  df-atl 40275  df-cvlat 40299  df-hlat 40328  df-llines 40475  df-lplanes 40476  df-lvols 40477  df-lines 40478  df-psubsp 40480  df-pmap 40481  df-padd 40773  df-lhyp 40965  df-laut 40966  df-ldil 41081  df-ltrn 41082

This theorem is used by:  cdleme51finvtrN  41535  cdleme50ex  41536  cdlemg1a  41547  cdlemg1ltrnlem  41551