cdlemk35u

Description: Substitution version of cdlemk35 . (Contributed by NM, 31-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	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
	cdlemk5.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
Assertion	cdlemk35u	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G) \in T$

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		cdlemk5.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
13		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F = N$
14		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to G \in T$
15	11 12	cdlemk40t	$⊢ (F = N \land G \in T) \to U (G) = G$
16	13 14 15	syl2anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G) = G$
17	16 14	eqeltrd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G) \in T$
18		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \neq N$
19		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to G \in T$
20	11 12	cdlemk40f	$⊢ (F \neq N \land G \in T) \to U (G) = ⦋ G / g ⦌ X$
21	18 19 20	syl2anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to U (G) = ⦋ G / g ⦌ X$
22		simpl1l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (K \in HL \land W \in H)$
23		simpl21	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \in T$
24		simpl22	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to N \in T$
25		simpl1r	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to R (F) = R (N)$
26	1 6 7 8	trlnid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (F \neq N \land R (F) = R (N))) \to F \neq {I ↾}_{B}$
27	22 23 24 18 25 26	syl122anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \neq {I ↾}_{B}$
28	23 27	jca	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (F \in T \land F \neq {I ↾}_{B})$
29		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
30	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
31	22 28 19 24 29 25 30	syl132anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to ⦋ G / g ⦌ X \in T$
32	21 31	eqeltrd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to U (G) \in T$
33	17 32	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G) \in T$

Metamath Proof Explorer

Theorem cdlemk35u

Proof