cdlemk45

Description: Part of proof of Lemma K of Crawley p. 118. Line 37, p. 119. G , I stand for g, h. X represents tau. They do not explicitly mention the requirement ` ( G o. I ) =/= ( _I |`B ) . (Contributed by NM, 22-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk5.b	$⊢ B = {Base}_{K}$
	cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk5.a	$⊢ A = Atoms (K)$
	cdlemk5.h	$⊢ H = LHyp (K)$
	cdlemk5.t	$⊢ T = LTrn (K) (W)$
	cdlemk5.r	$⊢ R = trL (K) (W)$
	cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
Assertion	cdlemk45	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk5.b	$⊢ B = {Base}_{K}$
2		cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk5.a	$⊢ A = Atoms (K)$
6		cdlemk5.h	$⊢ H = LHyp (K)$
7		cdlemk5.t	$⊢ T = LTrn (K) (W)$
8		cdlemk5.r	$⊢ R = trL (K) (W)$
9		cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
11		cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
12		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (K \in HL \land W \in H)$
13		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (F \in T \land F \neq {I ↾}_{B})$
14		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to G \in T$
15		simp31	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to I \in T$
16	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
17	12 14 15 16	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to G \circ I \in T$
18		simp33	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to G \circ I \neq {I ↾}_{B}$
19	17 18	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (G \circ I \in T \land G \circ I \neq {I ↾}_{B})$
20		simp2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
21		simp32	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to I \neq {I ↾}_{B}$
22	1 2 3 4 5 6 7 8 9 10 11	cdlemk11t	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \circ I \in T \land G \circ I \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ {(G \circ I)}^{-1}))$
23	12 13 19 20 15 21 22	syl312anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ {(G \circ I)}^{-1}))$
24		cnvco	$⊢ {(G \circ I)}^{-1} = I^{-1} \circ G^{-1}$
25	24	coeq2i	$⊢ I \circ {(G \circ I)}^{-1} = I \circ (I^{-1} \circ G^{-1})$
26		coass	$⊢ (I \circ I^{-1}) \circ G^{-1} = I \circ (I^{-1} \circ G^{-1})$
27	25 26	eqtr4i	$⊢ I \circ {(G \circ I)}^{-1} = (I \circ I^{-1}) \circ G^{-1}$
28	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land I \in T) \to I : B \underset{1-1 onto}{⟶} B$
29	12 15 28	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to I : B \underset{1-1 onto}{⟶} B$
30		f1ococnv2	$⊢ I : B \underset{1-1 onto}{⟶} B \to I \circ I^{-1} = {I ↾}_{B}$
31	29 30	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to I \circ I^{-1} = {I ↾}_{B}$
32	31	coeq1d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (I \circ I^{-1}) \circ G^{-1} = ({I ↾}_{B}) \circ G^{-1}$
33	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land G \in T) \to G : B \underset{1-1 onto}{⟶} B$
34	12 14 33	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to G : B \underset{1-1 onto}{⟶} B$
35		f1ocnv	$⊢ G : B \underset{1-1 onto}{⟶} B \to G^{-1} : B \underset{1-1 onto}{⟶} B$
36		f1of	$⊢ G^{-1} : B \underset{1-1 onto}{⟶} B \to G^{-1} : B ⟶ B$
37		fcoi2	$⊢ G^{-1} : B ⟶ B \to ({I ↾}_{B}) \circ G^{-1} = G^{-1}$
38	34 35 36 37	4syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ({I ↾}_{B}) \circ G^{-1} = G^{-1}$
39	32 38	eqtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to (I \circ I^{-1}) \circ G^{-1} = G^{-1}$
40	27 39	eqtrid	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to I \circ {(G \circ I)}^{-1} = G^{-1}$
41	40	fveq2d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to R (I \circ {(G \circ I)}^{-1}) = R (G^{-1})$
42	6 7 8	trlcnv	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G^{-1}) = R (G)$
43	12 14 42	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to R (G^{-1}) = R (G)$
44	41 43	eqtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to R (I \circ {(G \circ I)}^{-1}) = R (G)$
45	44	oveq2d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ {(G \circ I)}^{-1}) = ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)$
46	23 45	breqtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$

Metamath Proof Explorer

Theorem cdlemk45

Proof