trlcone - Mathbox for Norm Megill

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Theorem trlcone 40730

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 1229	. . 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 1204	. . . . . . . . 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 1287	. . . . . . . . . 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 40148	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
7	2, 3, 6	syl2anc 584	. . . . . . . . 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 1288	. . . . . . . . . 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 40721	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
10	2, 3, 8, 9	syl3anc 1373	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
12		eqid 2737	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
13		trlcone.r	. . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	11, 12, 4, 5, 13	trlco 40729	. . . . . . . . 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 1373	. . . . . . . 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 40126	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
18	2, 3, 17	syl2anc 584	. . . . . . . . . . . . 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 6877	. . . . . . . . . . . . 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 5872	. . . . . . . . . . 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 40126	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵)
23	2, 8, 22	syl2anc 584	. . . . . . . . . . . 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 6848	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵)
25		fcoi2 6783	. . . . . . . . . . . 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 2777	. . . . . . . . . 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 6285	. . . . . . . . . 10 ⊢ ((^◡𝐹 ∘ 𝐹) ∘ 𝐺) = (^◡𝐹 ∘ (𝐹 ∘ 𝐺))
29	27, 28	eqtr3di 2792	. . . . . . . . 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 6910	. . . . . . . 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 1285	. . . . . . . . . 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 1138	. . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
34	12, 33	hlatjidm 39370	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ (Atoms‘𝐾)) → ((𝑅‘𝐹)(join‘𝐾)(𝑅‘𝐹)) = (𝑅‘𝐹))
35	31, 32, 34	syl2anc 584	. . . . . . . . 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 40167	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
37	2, 3, 36	syl2anc 584	. . . . . . . . . . 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 2743	. . . . . . . . . 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 1139	. . . . . . . . . 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 7449	. . . . . . . . 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 2779	. . . . . . . 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 5175	. . . . . . 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 39361	. . . . . . . . 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 1290	. . . . . . . . 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 40175	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
47	2, 8, 45, 46	syl3anc 1373	. . . . . . . 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 39312	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 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 2743	. . . . 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 1122	. . . 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 2961	. . 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 1230	. . . . 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 1192	. . . . . . 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 1228	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑅‘𝐹) = (0.‘𝐾)) → 𝐺 ∈ 𝑇)
58		eqid 2737	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
59	16, 58, 4, 5, 13	trlid0b 40180	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ 𝐵) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
60	56, 57, 59	syl2anc 584	. . . . . 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 2985	. . . . 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 232	. . . 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 2996	. . 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 484	. . 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 1227	. . . . . . . 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 40180	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = (0.‘𝐾)))
67	56, 65, 66	syl2anc 584	. . . . . . 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 257	. . . . . 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 5872	. . . . 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 584	. . . . . 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 2777	. . . 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 6910	. . 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 3018	. 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 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
76		simp2l 1200	. . 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 40173	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∨ (𝑅‘𝐹) = (0.‘𝐾)))
78	75, 76, 77	syl2anc 584	. 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 961	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 I cid 5577 ^◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 0.cp0 18468 Atomscatm 39264 AtLatcal 39265 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161

This theorem is referenced by: trljco 40742 cdlemh2 40818 cdlemh 40819 cdlemk3 40835 cdlemk12 40852 cdlemk12u 40874 cdlemkfid1N 40923 cdlemk54 40960

W3C validator