Theorem cdleme43cN 41063

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 last line: r v g(s) = r v v₂. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme43.b	⊢ 𝐵 = (Base‘𝐾)
cdleme43.l	⊢ ≤ = (le‘𝐾)
cdleme43.j	⊢ ∨ = (join‘𝐾)
cdleme43.m	⊢ ∧ = (meet‘𝐾)
cdleme43.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme43.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme43.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme43.x	⊢ 𝑋 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdleme43.c	⊢ 𝐶 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme43.f	⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐶 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
cdleme43.d	⊢ 𝐷 = ((𝑆 ∨ 𝑋) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊)))
cdleme43.g	⊢ 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)))
cdleme43.e	⊢ 𝐸 = ((𝐷 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝐷) ∧ 𝑊)))
cdleme43.v	⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊)
cdleme43.y	⊢ 𝑌 = ((𝑅 ∨ 𝐷) ∧ 𝑊)

Assertion

Ref	Expression
cdleme43cN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ 𝐷) = (𝑅 ∨ 𝑌))

Proof of Theorem cdleme43cN

Step	Hyp	Ref	Expression
1		simp11 1213	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp22 1217	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
3		simp1 1145	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
4		simp21 1216	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
5		simp23 1218	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
6		simp3 1147	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
7		cdleme43.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
8		cdleme43.l	. . . 4 ⊢ ≤ = (le‘𝐾)
9		cdleme43.j	. . . 4 ⊢ ∨ = (join‘𝐾)
10		cdleme43.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
11		cdleme43.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdleme43.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdleme43.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
14		cdleme43.x	. . . 4 ⊢ 𝑋 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
15		cdleme43.c	. . . 4 ⊢ 𝐶 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
16		cdleme43.f	. . . 4 ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝐶 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
17		cdleme43.d	. . . 4 ⊢ 𝐷 = ((𝑆 ∨ 𝑋) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑆) ∧ 𝑊)))
18		cdleme43.g	. . . 4 ⊢ 𝐺 = ((𝑄 ∨ 𝑃) ∧ (𝐷 ∨ ((𝑍 ∨ 𝑆) ∧ 𝑊)))
19		cdleme43.e	. . . 4 ⊢ 𝐸 = ((𝐷 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝐷) ∧ 𝑊)))
20		cdleme43.v	. . . 4 ⊢ 𝑉 = ((𝑍 ∨ 𝑆) ∧ 𝑊)
21		cdleme43.y	. . . 4 ⊢ 𝑌 = ((𝑅 ∨ 𝐷) ∧ 𝑊)
22	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdleme43bN 41062	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐷 ∈ 𝐴 ∧ ¬ 𝐷 ≤ 𝑊))
23	3, 4, 5, 6, 22	syl121anc 1390	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐷 ∈ 𝐴 ∧ ¬ 𝐷 ≤ 𝑊))
24	7, 8, 9, 10, 11, 12, 21	cdleme42a 41043	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝐷 ∈ 𝐴 ∧ ¬ 𝐷 ≤ 𝑊)) → (𝑅 ∨ 𝐷) = (𝑅 ∨ 𝑌))
25	1, 2, 23, 24	syl3anc 1386	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ 𝐷) = (𝑅 ∨ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951   class class class wbr 5094  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  lecple 17269  joincjn 18319  meetcmee 18320  Atomscatm 39835  HLchlt 39922  LHypclh 40556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-proset 18302  df-poset 18321  df-plt 18336  df-lub 18352  df-glb 18353  df-join 18354  df-meet 18355  df-p0 18431  df-p1 18432  df-lat 18440  df-clat 18507  df-oposet 39748  df-ol 39750  df-oml 39751  df-covers 39838  df-ats 39839  df-atl 39870  df-cvlat 39894  df-hlat 39923  df-lines 40073  df-psubsp 40075  df-pmap 40076  df-padd 40368  df-lhyp 40560

This theorem is referenced by:  cdlemeg46rjgN  41094