Theorem cdlemg7N 41460

Description: TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg7.b	⊢ 𝐵 = (Base‘𝐾)
cdlemg7.l	⊢ ≤ = (le‘𝐾)
cdlemg7.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg7.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg7.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg7N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)

Proof of Theorem cdlemg7N

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl31 1273	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → 𝐹 ∈ 𝑇)
3		simpl32 1274	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → 𝐺 ∈ 𝑇)
4		simpl2r 1246	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → 𝑋 ∈ 𝐵)
5		cdlemg7.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		cdlemg7.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg7.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	5, 6, 7	ltrncl 40959	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) ∈ 𝐵)
9	1, 3, 4, 8	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐺‘𝑋) ∈ 𝐵)
10		simpr 490	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → 𝑋 ≤ 𝑊)
11		cdlemg7.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
12	5, 11, 6, 7	ltrnval1 40968	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐺‘𝑋) = 𝑋)
13	1, 3, 4, 10, 12	syl112anc 1401	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐺‘𝑋) = 𝑋)
14	13, 10	eqbrtrd 5135	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐺‘𝑋) ≤ 𝑊)
15	5, 11, 6, 7	ltrnval1 40968	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑋) ∈ 𝐵 ∧ (𝐺‘𝑋) ≤ 𝑊)) → (𝐹‘(𝐺‘𝑋)) = (𝐺‘𝑋))
16	1, 2, 9, 14, 15	syl112anc 1401	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐹‘(𝐺‘𝑋)) = (𝐺‘𝑋))
17	16, 13	eqtrd 2800	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑋 ≤ 𝑊) → (𝐹‘(𝐺‘𝑋)) = 𝑋)
18		simpl1 1210	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
19		simpl2l 1245	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
20		simpl2r 1246	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → 𝑋 ∈ 𝐵)
21		simpr 490	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → ¬ 𝑋 ≤ 𝑊)
22	20, 21	jca 521	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
23		simpl31 1273	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → 𝐹 ∈ 𝑇)
24		simpl32 1274	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → 𝐺 ∈ 𝑇)
25		simpl33 1275	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → (𝐹‘(𝐺‘𝑃)) = 𝑃)
26		cdlemg7.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
27	5, 11, 26, 6, 7	cdlemg7aN 41459	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)
28	18, 19, 22, 23, 24, 25, 27	syl123anc 1414	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ¬ 𝑋 ≤ 𝑊) → (𝐹‘(𝐺‘𝑋)) = 𝑋)
29	17, 28	pm2.61dan 825	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ‘cfv 6540  Basecbs 17293  lecple 17341  Atomscatm 40097  HLchlt 40184  LHypclh 40818  LTrncltrn 40935

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 39787

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 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-p1 18504  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-llines 40332  df-lplanes 40333  df-lvols 40334  df-lines 40335  df-psubsp 40337  df-pmap 40338  df-padd 40630  df-lhyp 40822  df-laut 40823  df-ldil 40938  df-ltrn 40939  df-trl 40993

This theorem is used by: (None)