cdlemk34

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 18-Jul-2013)

	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	cdlemk34	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$

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		simpll1	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (K \in HL \land W \in H)$
14		simplr1	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z \in T$
15		simpl1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (K \in HL \land W \in H)$
16		simpl3r	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (F) = R (N)$
17		simp22l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \in T$
18	17	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
19	15 16 18	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T)$
20	19	adantr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to ((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T)$
21		simp21l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to F \in T$
22	21	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \in T$
23		simpr2	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \in T$
24		simpl23	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to N \in T$
25	22 23 24	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land b \in T \land N \in T)$
26	25	adantr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (F \in T \land b \in T \land N \in T)$
27		simpr32	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (F)$
28		simpr33	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (G)$
29	27 28	jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (R (b) \neq R (F) \land R (b) \neq R (G))$
30	29	adantr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (R (b) \neq R (F) \land R (b) \neq R (G))$
31		simp21r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to F \neq {I ↾}_{B}$
32	31	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \neq {I ↾}_{B}$
33		simp22r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \neq {I ↾}_{B}$
34	33	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \neq {I ↾}_{B}$
35		simpr31	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \neq {I ↾}_{B}$
36	32 34 35	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
37	36	adantr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
38		simpl3l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
39	38	adantr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
40	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land b \in T \land N \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b Y G \in T$
41	20 26 30 37 39 40	syl113anc	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to b Y G \in T$
42		simpr	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z (P) = (b Y G) (P)$
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	13 14 41 39 42 43	syl311anc	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \land z (P) = (b Y G) (P)) \to z = b Y G$
45	44	ex	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z (P) = (b Y G) (P) \to z = b Y G)$
46	12 45	impbid2	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z = b Y G \leftrightarrow z (P) = (b Y G) (P))$
47		simp1	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (K \in HL \land W \in H)$
48		simp3r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (F) = R (N)$
49	47 48	jca	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
50	49	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
51	32 35 34	3jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
52	1 2 3 4 5 6 7 8 9 10	cdlemk32	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (b Y G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))$
53	50 25 18 29 51 38 52	syl123anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (b Y G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))$
54	53	eqeq2d	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z (P) = (b Y G) (P) \leftrightarrow z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1})))$
55	46 54	bitrd	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (z \in T \land b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (z = b Y G \leftrightarrow z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1})))$
56	55	3exp2	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (z \in T \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to (z = b Y G \leftrightarrow z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))))$
57	56	imp31	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land z \in T) \land b \in T) \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to (z = b Y G \leftrightarrow z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$
58	57	pm5.74d	$⊢ ((((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land z \in T) \land b \in T) \to (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G) \leftrightarrow ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$
59	58	ralbidva	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land z \in T) \to (\forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G) \leftrightarrow \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$
60	59	riotabidva	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z = b Y G)) = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$
61	11 60	syl5eq	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (((P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))) \underset{˙}{\lor} R (G \circ b^{-1}))))$

Metamath Proof Explorer

Theorem cdlemk34

Proof