cdlemk39s-id

Description: Substitution version of cdlemk39 with non-identity requirement on G removed. TODO: Can any commonality with cdlemk35s be exploited? (Contributed by NM, 26-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	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
Assertion	cdlemk39s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$

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		simpl1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to (K \in HL \land W \in H)$
13		simp21l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to F \in T$
14		simp23	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to N \in T$
15		simp3r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (F) = R (N)$
16	13 14 15	3jca	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (F \in T \land N \in T \land R (F) = R (N))$
17	16	adantr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to (F \in T \land N \in T \land R (F) = R (N))$
18		simpl3l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simpr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to G = {I ↾}_{B}$
20	1 2 3 4 5 6 7 8 9 10 11	cdlemkid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T \land R (F) = R (N)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G = {I ↾}_{B})) \to ⦋ G / g ⦌ X = {I ↾}_{B}$
21	12 17 18 19 20	syl112anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to ⦋ G / g ⦌ X = {I ↾}_{B}$
22	21	fveq2d	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to R (⦋ G / g ⦌ X) = R ({I ↾}_{B})$
23		simpl1l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to K \in HL$
24		simpl1r	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to W \in H$
25		eqid	$⊢ 0. (K) = 0. (K)$
26	1 25 6 8	trlid0	$⊢ (K \in HL \land W \in H) \to R ({I ↾}_{B}) = 0. (K)$
27	23 24 26	syl2anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to R ({I ↾}_{B}) = 0. (K)$
28	22 27	eqtrd	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to R (⦋ G / g ⦌ X) = 0. (K)$
29		hlop	$⊢ K \in HL \to K \in OP$
30	23 29	syl	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to K \in OP$
31		simpl22	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to G \in T$
32	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \in B$
33	12 31 32	syl2anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to R (G) \in B$
34	1 2 25	op0le	$⊢ (K \in OP \land R (G) \in B) \to 0. (K) \underset{˙}{\leq} R (G)$
35	30 33 34	syl2anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to 0. (K) \underset{˙}{\leq} R (G)$
36	28 35	eqbrtrd	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G = {I ↾}_{B}) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$
37		simpl1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to (K \in HL \land W \in H)$
38		simpl21	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to (F \in T \land F \neq {I ↾}_{B})$
39		simpl22	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to G \in T$
40		simpr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to G \neq {I ↾}_{B}$
41	39 40	jca	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to (G \in T \land G \neq {I ↾}_{B})$
42		simpl23	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to N \in T$
43		simpl3	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
44	1 2 3 4 5 6 7 8 9 10 11	cdlemk39s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$
45	37 38 41 42 43 44	syl131anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land G \neq {I ↾}_{B}) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$
46	36 45	pm2.61dane	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$

Metamath Proof Explorer

Theorem cdlemk39s-id

Proof