cdlemg11b - Mathbox for Norm Megill

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Theorem cdlemg11b 41147

Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg8.l	⊢ ≤ = (le‘𝐾)
cdlemg8.j	⊢ ∨ = (join‘𝐾)
cdlemg8.m	⊢ ∧ = (meet‘𝐾)
cdlemg8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg10.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemg11b	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) ≠ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))

**Proof of Theorem cdlemg11b**
Step	Hyp	Ref	Expression
1		simp33 1219	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
2		simpl1 1199	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simpl31 1262	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝐺 ∈ 𝑇)
4		simpl2l 1234	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5		cdlemg8.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6		cdlemg8.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
7		cdlemg8.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
8		cdlemg8.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemg8.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdlemg8.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		cdlemg10.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12	5, 6, 7, 8, 9, 10, 11	trlval2 40668	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
13	2, 3, 4, 12	syl3anc 1380	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
14		eqid 2741	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
15		simpl1l 1232	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝐾 ∈ HL)
16	15	hllatd 39869	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝐾 ∈ Lat)
17		simp2ll 1248	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
18	17	adantr 482	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑃 ∈ 𝐴)
19	14, 8	atbase 39794	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑃 ∈ (Base‘𝐾))
21	14, 9, 10	ltrncl 40630	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐺‘𝑃) ∈ (Base‘𝐾))
22	2, 3, 20, 21	syl3anc 1380	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝐺‘𝑃) ∈ (Base‘𝐾))
23	14, 6	latjcl 18400	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐺‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
24	16, 20, 22, 23	syl3anc 1380	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
25		simpl1r 1233	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑊 ∈ 𝐻)
26	14, 9	lhpbase 40503	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
27	25, 26	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑊 ∈ (Base‘𝐾))
28	14, 7	latmcl 18401	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
29	16, 24, 27, 28	syl3anc 1380	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
30		simpl2r 1235	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑄 ∈ 𝐴)
31	14, 8	atbase 39794	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑄 ∈ (Base‘𝐾))
33	14, 6	latjcl 18400	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
34	16, 20, 32, 33	syl3anc 1380	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
35	14, 5, 7	latmle1 18425	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ (𝐺‘𝑃)))
36	16, 24, 27, 35	syl3anc 1380	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ (𝐺‘𝑃)))
37	14, 5, 6	latlej1 18409	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑄))
38	16, 20, 32, 37	syl3anc 1380	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → 𝑃 ≤ (𝑃 ∨ 𝑄))
39	14, 9, 10	ltrncl 40630	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
40	2, 3, 32, 39	syl3anc 1380	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝐺‘𝑄) ∈ (Base‘𝐾))
41	14, 5, 6	latlej1 18409	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝐺‘𝑃) ∈ (Base‘𝐾) ∧ (𝐺‘𝑄) ∈ (Base‘𝐾)) → (𝐺‘𝑃) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
42	16, 22, 40, 41	syl3anc 1380	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝐺‘𝑃) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
43		simpr 486	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
44	42, 43	breqtrrd 5102	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄))
45	14, 5, 6	latjle12 18411	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺‘𝑃) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ (𝐺‘𝑃)) ≤ (𝑃 ∨ 𝑄)))
46	16, 20, 22, 34, 45	syl13anc 1381	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ (𝐺‘𝑃)) ≤ (𝑃 ∨ 𝑄)))
47	38, 44, 46	mpbi2and 719	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) ≤ (𝑃 ∨ 𝑄))
48	14, 5, 16, 29, 24, 34, 36, 47	lattrd 18407	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
49	13, 48	eqbrtrd 5096	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) → (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
50	49	ex 414	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) → (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄)))
51	50	necon3bd 2950	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) → (𝑃 ∨ 𝑄) ≠ ((𝐺‘𝑃) ∨ (𝐺‘𝑄))))
52	1, 51	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) ≠ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 class class class wbr 5074 ‘cfv 6488 (class class class)co 7359 Basecbs 17174 lecple 17222 joincjn 18272 meetcmee 18273 Latclat 18392 Atomscatm 39768 HLchlt 39855 LHypclh 40489 LTrncltrn 40606 trLctrl 40663

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 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 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 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-map 8769 df-poset 18274 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-lat 18393 df-ats 39772 df-atl 39803 df-cvlat 39827 df-hlat 39856 df-lhyp 40493 df-laut 40494 df-ldil 40609 df-ltrn 40610 df-trl 40664

This theorem is referenced by: cdlemg12b 41149

W3C validator