cdlemkfid1N - Mathbox for Norm Megill

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Theorem cdlemkfid1N 40526

Description: Lemma for cdlemkfid3N 40530. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cdlemk5.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk5.l	⊢ ≤ = (le‘𝐾)
cdlemk5.j	⊢ ∨ = (join‘𝐾)
cdlemk5.m	⊢ ∧ = (meet‘𝐾)
cdlemk5.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk5.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk5.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk5.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemkfid1N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝐺‘𝑃))

**Proof of Theorem cdlemkfid1N**
Step	Hyp	Ref	Expression
1		simp1 1133	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp23 1205	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐺 ∈ 𝑇)
3		simp3r 1199	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		cdlemk5.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		cdlemk5.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
6		cdlemk5.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk5.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk5.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		cdlemk5.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
10	4, 5, 6, 7, 8, 9	trljat3 39773	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐺)) = ((𝐺‘𝑃) ∨ (𝑅‘𝐺)))
11	1, 2, 3, 10	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑅‘𝐺)) = ((𝐺‘𝑃) ∨ (𝑅‘𝐺)))
12		simp1l 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐾 ∈ HL)
13		simp21 1203	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐹 ∈ 𝑇)
14		simp3rl 1243	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
15	4, 6, 7, 8	ltrnat 39745	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
16	1, 13, 14, 15	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐹‘𝑃) ∈ 𝐴)
17	4, 6, 7, 8	ltrnat 39745	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
18	1, 2, 14, 17	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐺‘𝑃) ∈ 𝐴)
19	5, 6	hlatjcom 38972	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ (𝐹‘𝑃)))
20	12, 16, 18, 19	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ (𝐹‘𝑃)))
21	4, 5, 6, 7, 8, 9	trlcoabs2N 40327	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
22	1, 13, 2, 3, 21	syl121anc 1372	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
23	7, 8, 9	trlcocnv 40325	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
24	1, 13, 2, 23	syl3anc 1368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
25	24	oveq2d 7435	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐺))) = ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
26	4, 5, 6, 7, 8, 9	trlcoabs2N 40327	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐺))) = ((𝐺‘𝑃) ∨ (𝐹‘𝑃)))
27	1, 2, 13, 3, 26	syl121anc 1372	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐺))) = ((𝐺‘𝑃) ∨ (𝐹‘𝑃)))
28	25, 27	eqtr3d 2767	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐺‘𝑃) ∨ (𝐹‘𝑃)))
29	20, 22, 28	3eqtr4d 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
30	11, 29	oveq12d 7437	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (((𝐺‘𝑃) ∨ (𝑅‘𝐺)) ∧ ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
31		cdlemk5.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
32	31, 7, 8, 9	trlcl 39769	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ 𝐵)
33	1, 2, 32	syl2anc 582	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐺) ∈ 𝐵)
34		simp1r 1195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
35		simp3l 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐺) ≠ (𝑅‘𝐹))
36	6, 7, 8, 9	trlcocnvat 40329	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
37	12, 34, 2, 13, 35, 36	syl221anc 1378	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
38	4, 6, 7, 8	ltrnel 39744	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
39	1, 2, 3, 38	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
40	7, 8	ltrncnv 39751	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
41	1, 13, 40	syl2anc 582	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ^◡𝐹 ∈ 𝑇)
42	7, 8, 9	trlcnv 39770	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
43	1, 13, 42	syl2anc 582	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
44	35, 43	neeqtrrd 3004	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐺) ≠ (𝑅‘^◡𝐹))
45		simp22 1204	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐹 ≠ ( I ↾ 𝐵))
46	31, 7, 8	ltrncnvnid 39732	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ^◡𝐹 ≠ ( I ↾ 𝐵))
47	1, 13, 45, 46	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ^◡𝐹 ≠ ( I ↾ 𝐵))
48	31, 7, 8, 9	trlcone 40333	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘^◡𝐹) ∧ ^◡𝐹 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
49	1, 2, 41, 44, 47, 48	syl122anc 1376	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
50		eqid 2725	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
51	50, 6, 7, 8, 9	trlator0 39776	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾)))
52	1, 2, 51	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾)))
53	4, 7, 8, 9	trlle 39789	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
54	12, 34, 2, 53	syl21anc 836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐺) ≤ 𝑊)
55	7, 8	ltrnco 40324	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
56	1, 2, 41, 55	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
57	4, 7, 8, 9	trlle 39789	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
58	1, 56, 57	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
59	4, 5, 50, 6, 7	lhp2at0nle 39640	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾)) ∧ (𝑅‘𝐺) ≤ 𝑊) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)) → ¬ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ ((𝐺‘𝑃) ∨ (𝑅‘𝐺)))
60	1, 39, 49, 52, 54, 37, 58, 59	syl322anc 1395	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ¬ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ ((𝐺‘𝑃) ∨ (𝑅‘𝐺)))
61		cdlemk5.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
62	31, 4, 5, 61, 6	2llnma1b 39391	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐺) ∈ 𝐵 ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) ∧ ¬ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ ((𝐺‘𝑃) ∨ (𝑅‘𝐺))) → (((𝐺‘𝑃) ∨ (𝑅‘𝐺)) ∧ ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝐺‘𝑃))
63	12, 33, 18, 37, 60, 62	syl131anc 1380	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (((𝐺‘𝑃) ∨ (𝑅‘𝐺)) ∧ ((𝐺‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝐺‘𝑃))
64	30, 63	eqtrd 2765	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝐺‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 I cid 5575 ^◡ccnv 5677 ↾ cres 5680 ∘ ccom 5682 ‘cfv 6549 (class class class)co 7419 Basecbs 17188 lecple 17248 joincjn 18311 meetcmee 18312 0.cp0 18423 Atomscatm 38867 HLchlt 38954 LHypclh 39589 LTrncltrn 39706 trLctrl 39763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-riotaBAD 38557

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-undef 8279 df-map 8847 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764

This theorem is referenced by: cdlemkfid2N 40528 cdlemkfid3N 40530

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