cdlemk42

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 20-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	cdlemk42	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y$

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		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
13	1 2 3 4 5 6 7 8 9 10 11	cdlemk36	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \to X (P) = Y$
14	13	sbcth	$⊢ G \in T \to \underset{˙}{[} G / g \underset{˙}{]} ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \to X (P) = Y)$
15		sbcimg	$⊢ G \in T \to (\underset{˙}{[} G / g \underset{˙}{]} ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \to X (P) = Y) \leftrightarrow (\underset{˙}{[} G / g \underset{˙}{]} (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \to \underset{˙}{[} G / g \underset{˙}{]} X (P) = Y))$
16	14 15	mpbid	$⊢ G \in T \to (\underset{˙}{[} G / g \underset{˙}{]} (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \to \underset{˙}{[} G / g \underset{˙}{]} X (P) = Y)$
17		eleq1	$⊢ g = G \to (g \in T \leftrightarrow G \in T)$
18		neeq1	$⊢ g = G \to (g \neq {I ↾}_{B} \leftrightarrow G \neq {I ↾}_{B})$
19	17 18	anbi12d	$⊢ g = G \to ((g \in T \land g \neq {I ↾}_{B}) \leftrightarrow (G \in T \land G \neq {I ↾}_{B}))$
20	19	3anbi3d	$⊢ g = G \to (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \leftrightarrow ((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})))$
21		fveq2	$⊢ g = G \to R (g) = R (G)$
22	21	neeq2d	$⊢ g = G \to (R (b) \neq R (g) \leftrightarrow R (b) \neq R (G))$
23	22	3anbi3d	$⊢ g = G \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \leftrightarrow (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))$
24	23	anbi2d	$⊢ g = G \to ((b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g))) \leftrightarrow (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))))$
25	20 24	3anbi13d	$⊢ g = G \to ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \leftrightarrow (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))))$
26	25	sbcieg	$⊢ G \in T \to (\underset{˙}{[} G / g \underset{˙}{]} (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (g \in T \land g \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)))) \leftrightarrow (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))))$
27		sbceqg	$⊢ G \in T \to (\underset{˙}{[} G / g \underset{˙}{]} X (P) = Y \leftrightarrow ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y)$
28		csbfv12	$⊢ ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ X (⦋ G / g ⦌ P)$
29		csbconstg	$⊢ G \in T \to ⦋ G / g ⦌ P = P$
30	29	fveq2d	$⊢ G \in T \to ⦋ G / g ⦌ X (⦋ G / g ⦌ P) = ⦋ G / g ⦌ X (P)$
31	28 30	syl5eq	$⊢ G \in T \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ X (P)$
32	31	eqeq1d	$⊢ G \in T \to (⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y \leftrightarrow ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y)$
33	27 32	bitrd	$⊢ G \in T \to (\underset{˙}{[} G / g \underset{˙}{]} X (P) = Y \leftrightarrow ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y)$
34	16 26 33	3imtr3d	$⊢ G \in T \to ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y)$
35	12 34	mpcom	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y$

Metamath Proof Explorer

Theorem cdlemk42

Proof