Theorem cdlemg7aN 41132

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
cdlemg7aN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)

Proof of Theorem cdlemg7aN

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1205	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐾 ∈ HL)
2		simp1r 1206	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑊 ∈ 𝐻)
3		simp2r 1208	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
4		cdlemg7.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
5		cdlemg7.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		eqid 2741	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2741	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
8		cdlemg7.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemg7.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10	4, 5, 6, 7, 8, 9	lhpmcvr2 40531	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
11	1, 2, 3, 10	syl21anc 844	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
12		simp11 1211	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		simp2 1144	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑟 ∈ 𝐴)
14		simp3l 1209	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ 𝑟 ≤ 𝑊)
15	13, 14	jca 517	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
16		simp12r 1295	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
17		simp131 1316	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐹 ∈ 𝑇)
18		simp132 1317	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐺 ∈ 𝑇)
19		simp3r 1210	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
20		cdlemg7.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
21	4, 5, 6, 7, 8, 9, 20	cdlemg7fvN 41131	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑟))(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
22	12, 15, 16, 17, 18, 19, 21	syl123anc 1396	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑟))(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
23		simp12l 1294	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24		simp133 1318	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑃)) = 𝑃)
25	5, 8, 9, 20	cdlemg6 41130	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑟)) = 𝑟)
26	12, 23, 15, 17, 18, 24, 25	syl123anc 1396	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑟)) = 𝑟)
27	26	oveq1d 7375	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹‘(𝐺‘𝑟))(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
28	22, 27, 19	3eqtrd 2780	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ 𝑟 ∈ 𝐴 ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)
29	28	rexlimdv3a 3146	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → (𝐹‘(𝐺‘𝑋)) = 𝑋))
30	11, 29	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  lecple 17222  joincjn 18272  meetcmee 18273  Atomscatm 39770  HLchlt 39857  LHypclh 40491  LTrncltrn 40608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-riotaBAD 39460

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-undef 8217  df-map 8769  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18393  df-clat 18460  df-oposet 39683  df-ol 39685  df-oml 39686  df-covers 39773  df-ats 39774  df-atl 39805  df-cvlat 39829  df-hlat 39858  df-llines 40005  df-lplanes 40006  df-lvols 40007  df-lines 40008  df-psubsp 40010  df-pmap 40011  df-padd 40303  df-lhyp 40495  df-laut 40496  df-ldil 40611  df-ltrn 40612  df-trl 40666

This theorem is referenced by:  cdlemg7N  41133