cdlemk33N

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 . (Contributed by NM, 18-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
	cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk3.a	$⊢ A = Atoms (K)$
	cdlemk3.h	$⊢ H = LHyp (K)$
	cdlemk3.t	$⊢ T = LTrn (K) (W)$
	cdlemk3.r	$⊢ R = trL (K) (W)$
	cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
	cdlemk3.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G))$
Assertion	cdlemk33N	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (b Y G) (P)))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
11		cdlemk3.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G))$
12		fveq1	$⊢ z = b Y G \to z (P) = (b Y G) (P)$
13		simpl11	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to K \in HL$
14		simpl12	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to W \in H$
15	13 14	jca	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (K \in HL \land W \in H)$
16		simpl31	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z \in T$
17		simp11	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to K \in HL$
18		simp12	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to W \in H$
19	17 18	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (K \in HL \land W \in H)$
20		simp13	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (F) = R (N)$
21		simp22l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
22	19 20 21	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T)$
23	22	adantr	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to ((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T)$
24		simp211	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \in T$
25		simp32	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \in T$
26		simp213	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to N \in T$
27	24 25 26	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land b \in T \land N \in T)$
28	27	adantr	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (F \in T \land b \in T \land N \in T)$
29		simp332	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (F)$
30		simp333	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (G)$
31	29 30	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (R (b) \neq R (F) \land R (b) \neq R (G))$
32		simp212	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \neq {I ↾}_{B}$
33		simp22r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \neq {I ↾}_{B}$
34		simp331	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \neq {I ↾}_{B}$
35	32 33 34	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
36		simp23	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
37	31 35 36	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))$
38	37	adantr	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))$
39	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land b \in T \land N \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b Y G \in T$
40	23 28 38 39	syl3anc	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to b Y G \in T$
41		simpl23	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
42		simpr	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z (P) = (b Y G) (P)$
43	2 5 6 7	cdlemd	$⊢ (((K \in HL \land W \in H) \land z \in T \land b Y G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land z (P) = (b Y G) (P)) \to z = b Y G$
44	15 16 40 41 42 43	syl311anc	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z = b Y G$
45	44	ex	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z (P) = (b Y G) (P) \to z = b Y G)$
46	12 45	impbid2	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z = b Y G \leftrightarrow z (P) = (b Y G) (P))$
47	46	3expia	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))) \to (z = b Y G \leftrightarrow z (P) = (b Y G) (P)))$
48	47	3expd	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (z \in T \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to (z = b Y G \leftrightarrow z (P) = (b Y G) (P)))))$
49	48	imp31	$⊢ ((((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \land z \in T) \land b \in T) \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to (z = b Y G \leftrightarrow z (P) = (b Y G) (P)))$
50	49	pm5.74d	$⊢ ((((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \land z \in T) \land b \in T) \to (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G) \leftrightarrow ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (b Y G) (P)))$
51	50	ralbidva	$⊢ (((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \land z \in T) \to (\forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G) \leftrightarrow \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (b Y G) (P)))$
52	51	riotabidva	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G)) = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (b Y G) (P)))$
53	11 52	syl5eq	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (b Y G) (P)))$

Metamath Proof Explorer

Theorem cdlemk33N

Proof