cdlemk23-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the ( RC ) =/= ( RD ) requirement from cdlemk22-3 . (Contributed by NM, 7-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	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
Assertion	cdlemk23-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (D Y G) (P) = (C Y G) (P)$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (K \in HL \land W \in H)$
12		simp121	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to F \in T$
13		simp122	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to D \in T$
14		simp123	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to N \in T$
15		simp131	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to G \in T$
16		simp133	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to x \in T$
17	14 15 16	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (N \in T \land G \in T \land x \in T)$
18		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp221	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (F) = R (N)$
20		simp222	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to F \neq {I ↾}_{B}$
21		simp223	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to D \neq {I ↾}_{B}$
22		simp231	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to G \neq {I ↾}_{B}$
23	20 21 22	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
24		simp233	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to x \neq {I ↾}_{B}$
25		simp333	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (G) \neq R (x)$
26		simp332	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (x) \neq R (F)$
27	24 25 26	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (x \neq {I ↾}_{B} \land R (G) \neq R (x) \land R (x) \neq R (F))$
28		simp313	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (D) \neq R (F)$
29		simp32l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (G) \neq R (D)$
30		simp331	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (x) \neq R (D)$
31	28 29 30	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (R (D) \neq R (F) \land R (G) \neq R (D) \land R (x) \neq R (D))$
32	1 2 3 4 5 6 7 8 9 10	cdlemk22-3	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land x \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (x \neq {I ↾}_{B} \land R (G) \neq R (x) \land R (x) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (x) \neq R (D)))) \to (D Y G) (P) = (x Y G) (P)$
33	11 12 13 17 18 19 23 27 31 32	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (D Y G) (P) = (x Y G) (P)$
34		simp132	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to C \in T$
35		simp232	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to C \neq {I ↾}_{B}$
36	20 35 22	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
37		simp312	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (C) \neq R (F)$
38		simp311	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (G) \neq R (C)$
39		simp32r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to R (x) \neq R (C)$
40	37 38 39	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (R (C) \neq R (F) \land R (G) \neq R (C) \land R (x) \neq R (C))$
41	1 2 3 4 5 6 7 8 9 10	cdlemk22-3	$⊢ (((K \in HL \land W \in H) \land F \in T \land C \in T) \land ((N \in T \land G \in T \land x \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (x \neq {I ↾}_{B} \land R (G) \neq R (x) \land R (x) \neq R (F)) \land (R (C) \neq R (F) \land R (G) \neq R (C) \land R (x) \neq R (C)))) \to (C Y G) (P) = (x Y G) (P)$
42	11 12 34 17 18 19 36 27 40 41	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (C Y G) (P) = (x Y G) (P)$
43	33 42	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T \land x \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land x \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land (R (G) \neq R (D) \land R (x) \neq R (C)) \land (R (x) \neq R (D) \land R (x) \neq R (F) \land R (G) \neq R (x)))) \to (D Y G) (P) = (C Y G) (P)$

Metamath Proof Explorer

Theorem cdlemk23-3

Proof