trlcone - Mathbox for Norm Megill

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Theorem trlcone 41316

Description: If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)

**Hypotheses**
Ref	Expression
trlcone.b	⊢ 𝐵 = (Base‘𝐾)
trlcone.h	⊢ 𝐻 = (LHyp‘𝐾)
trlcone.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlcone.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
trlcone	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))

**Proof of Theorem trlcone**
Step	Hyp	Ref	Expression
1		simpl3l 1241	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
2		simp11 1216	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp12l 1299	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐹 ∈ 𝑇)
4		trlcone.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
5		trlcone.t	. . . . . . . . . . 11 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	4, 5	ltrncnv 40734	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
7	2, 3, 6	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ^◡𝐹 ∈ 𝑇)
8		simp12r 1300	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐺 ∈ 𝑇)
9	4, 5	ltrnco 41307	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
10	2, 3, 8, 9	syl3anc 1389	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝐹 ∘ 𝐺) ∈ 𝑇)
11		eqid 2761	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
12		eqid 2761	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
13		trlcone.r	. . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	11, 12, 4, 5, 13	trlco 41315	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ^◡𝐹 ∈ 𝑇 ∧ (𝐹 ∘ 𝐺) ∈ 𝑇) → (𝑅‘(^◡𝐹 ∘ (𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘^◡𝐹)(join‘𝐾)(𝑅‘(𝐹 ∘ 𝐺))))
15	2, 7, 10, 14	syl3anc 1389	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘(^◡𝐹 ∘ (𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘^◡𝐹)(join‘𝐾)(𝑅‘(𝐹 ∘ 𝐺))))
16		trlcone.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝐾)
17	16, 4, 5	ltrn1o 40712	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
18	2, 3, 17	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐹:𝐵–1-1-onto→𝐵)
19		f1ococnv1 6832	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
20	18, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
21	20	coeq1d 5831	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ((^◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
22	16, 4, 5	ltrn1o 40712	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
23	2, 8, 22	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐺:𝐵–1-1-onto→𝐵)
24		f1of 6802	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
25		fcoi2 6735	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
26	23, 24, 25	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
27	21, 26	eqtrd 2796	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ((^◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
28		coass 6249	. . . . . . . . . 10 ⊢ ((^◡𝐹 ∘ 𝐹) ∘ 𝐺) = (^◡𝐹 ∘ (𝐹 ∘ 𝐺))
29	27, 28	eqtr3di 2811	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐺 = (^◡𝐹 ∘ (𝐹 ∘ 𝐺)))
30	29	fveq2d 6867	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐺) = (𝑅‘(^◡𝐹 ∘ (𝐹 ∘ 𝐺))))
31		simp11l 1297	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐾 ∈ HL)
32		simp2 1149	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
33		eqid 2761	. . . . . . . . . . 11 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
34	12, 33	hlatjidm 39957	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝐹)(join‘𝐾)(𝑅‘𝐹)) = (𝑅‘𝐹))
35	31, 32, 34	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ((𝑅‘𝐹)(join‘𝐾)(𝑅‘𝐹)) = (𝑅‘𝐹))
36	4, 5, 13	trlcnv 40753	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
37	2, 3, 36	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
38	37	eqcomd 2767	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐹) = (𝑅‘^◡𝐹))
39		simp3 1150	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺)))
40	38, 39	oveq12d 7410	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ((𝑅‘𝐹)(join‘𝐾)(𝑅‘𝐹)) = ((𝑅‘^◡𝐹)(join‘𝐾)(𝑅‘(𝐹 ∘ 𝐺))))
41	35, 40	eqtr3d 2798	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐹) = ((𝑅‘^◡𝐹)(join‘𝐾)(𝑅‘(𝐹 ∘ 𝐺))))
42	15, 30, 41	3brtr4d 5131	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐺)(le‘𝐾)(𝑅‘𝐹))
43		hlatl 39948	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
44	31, 43	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐾 ∈ AtLat)
45		simp13r 1302	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → 𝐺 ≠ ( I ↾ 𝐵))
46	16, 33, 4, 5, 13	trlnidat 40761	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
47	2, 8, 45, 46	syl3anc 1389	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
48	11, 33	atcmp 39899	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ (𝑅‘𝐺) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝐺)(le‘𝐾)(𝑅‘𝐹) ↔ (𝑅‘𝐺) = (𝑅‘𝐹)))
49	44, 47, 32, 48	syl3anc 1389	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → ((𝑅‘𝐺)(le‘𝐾)(𝑅‘𝐹) ↔ (𝑅‘𝐺) = (𝑅‘𝐹)))
50	42, 49	mpbid 234	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐺) = (𝑅‘𝐹))
51	50	eqcomd 2767	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺))) → (𝑅‘𝐹) = (𝑅‘𝐺))
52	51	3expia 1133	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝐹) = (𝑅‘(𝐹 ∘ 𝐺)) → (𝑅‘𝐹) = (𝑅‘𝐺)))
53	52	necon3d 2977	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝐹) ≠ (𝑅‘𝐺) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺))))
54	1, 53	mpd 15	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))
55		simpl3r 1242	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐺 ≠ ( I ↾ 𝐵))
56		simpl1 1204	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
57		simpl2r 1240	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐺 ∈ 𝑇)
58		eqid 2761	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
59	16, 58, 4, 5, 13	trlid0b 40766	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ 𝐵) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
60	56, 57, 59	syl2anc 593	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐺 = ( I ↾ 𝐵) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
61	60	necon3bid 3000	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐺 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐺) ≠ (0.‘𝐾)))
62	55, 61	mpbid 234	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝐺) ≠ (0.‘𝐾))
63	62	necomd 3011	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (0.‘𝐾) ≠ (𝑅‘𝐺))
64		simpr 488	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝐹) = (0.‘𝐾))
65		simpl2l 1239	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐹 ∈ 𝑇)
66	16, 58, 4, 5, 13	trlid0b 40766	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = (0.‘𝐾)))
67	56, 65, 66	syl2anc 593	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = (0.‘𝐾)))
68	64, 67	mpbird 259	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐹 = ( I ↾ 𝐵))
69	68	coeq1d 5831	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐹 ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺))
70	56, 57, 22	syl2anc 593	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐺:𝐵–1-1-onto→𝐵)
71	70, 24, 25	3syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺)
72	69, 71	eqtrd 2796	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝐹 ∘ 𝐺) = 𝐺)
73	72	fveq2d 6867	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
74	63, 64, 73	3netr4d 3033	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))
75		simp1 1148	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
76		simp2l 1212	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → 𝐹 ∈ 𝑇)
77	58, 33, 4, 5, 13	trlator0 40759	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
78	75, 76, 77	syl2anc 593	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
79	54, 74, 78	mpjaodan 971	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5099 I cid 5539 ^◡ccnv 5644 ↾ cres 5647 ∘ ccom 5649 ⟶wf 6513 –1-1-onto→wf1o 6516 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 lecple 17276 joincjn 18326 0.cp0 18436 Atomscatm 39851 AtLatcal 39852 HLchlt 39938 LHypclh 40572 LTrncltrn 40689 trLctrl 40746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-undef 8248 df-map 8805 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747

This theorem is referenced by: trljco 41328 cdlemh2 41404 cdlemh 41405 cdlemk3 41421 cdlemk12 41438 cdlemk12u 41460 cdlemkfid1N 41509 cdlemk54 41546

W3C validator