cdlemk19u

Description: Part of Lemma K of Crawley p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with F , N , U . (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	cdlemk19u	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) = N$

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		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
14		simp1	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
15		simp2l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to F \in T$
16		simp2r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to N \in T$
17		simp3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk35u	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land F \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) \in T$
19	14 15 16 15 17 18	syl131anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) \in T$
20		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F = N$
21		simpl2l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F \in T$
22	11 12	cdlemk40t	$⊢ (F = N \land F \in T) \to U (F) = F$
23	20 21 22	syl2anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (F) = F$
24	23	fveq1d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (F) (P) = F (P)$
25		fveq1	$⊢ F = N \to F (P) = N (P)$
26	25	adantl	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F (P) = N (P)$
27	24 26	eqtrd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (F) (P) = N (P)$
28		simpl1	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
29		simpl2l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \in T$
30		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \neq N$
31		simpl2r	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to N \in T$
32		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
33	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk19u1	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq N \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) (P) = N (P)$
34	28 29 30 31 32 33	syl131anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to U (F) (P) = N (P)$
35	27 34	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) (P) = N (P)$
36	2 5 6 7	cdlemd	$⊢ (((K \in HL \land W \in H) \land U (F) \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land U (F) (P) = N (P)) \to U (F) = N$
37	13 19 16 17 35 36	syl311anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (F) = N$

Metamath Proof Explorer

Theorem cdlemk19u

Proof