mapdheq

Description: Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in Baer p. 45 line 24. (Contributed by NM, 4-Apr-2015)

	Ref	Expression
Hypotheses	mapdh.q	$⊢ Q = 0_{C}$
	mapdh.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
	mapdh.h	$⊢ H = LHyp (K)$
	mapdh.m	$⊢ M = mapd (K) (W)$
	mapdh.u	$⊢ U = DVecH (K) (W)$
	mapdh.v	$⊢ V = {Base}_{U}$
	mapdh.s	$⊢ \underset{˙}{-} = -_{U}$
	mapdhc.o	$⊢ \underset{˙}{0} = 0_{U}$
	mapdh.n	$⊢ N = LSpan (U)$
	mapdh.c	$⊢ C = LCDual (K) (W)$
	mapdh.d	$⊢ D = {Base}_{C}$
	mapdh.r	$⊢ R = -_{C}$
	mapdh.j	$⊢ J = LSpan (C)$
	mapdh.k	$⊢ φ \to (K \in HL \land W \in H)$
	mapdhc.f	$⊢ φ \to F \in D$
	mapdh.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdhcl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdhe.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdhe.g	$⊢ φ \to G \in D$
	mapdh.ne2	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
Assertion	mapdheq	$⊢ φ \to (I (⟨X, F, Y⟩) = G \leftrightarrow (M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})))$

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	$⊢ Q = 0_{C}$
2		mapdh.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
3		mapdh.h	$⊢ H = LHyp (K)$
4		mapdh.m	$⊢ M = mapd (K) (W)$
5		mapdh.u	$⊢ U = DVecH (K) (W)$
6		mapdh.v	$⊢ V = {Base}_{U}$
7		mapdh.s	$⊢ \underset{˙}{-} = -_{U}$
8		mapdhc.o	$⊢ \underset{˙}{0} = 0_{U}$
9		mapdh.n	$⊢ N = LSpan (U)$
10		mapdh.c	$⊢ C = LCDual (K) (W)$
11		mapdh.d	$⊢ D = {Base}_{C}$
12		mapdh.r	$⊢ R = -_{C}$
13		mapdh.j	$⊢ J = LSpan (C)$
14		mapdh.k	$⊢ φ \to (K \in HL \land W \in H)$
15		mapdhc.f	$⊢ φ \to F \in D$
16		mapdh.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdhcl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
18		mapdhe.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
19		mapdhe.g	$⊢ φ \to G \in D$
20		mapdh.ne2	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
21	1 2 17 15 18	mapdhval2	$⊢ φ \to I (⟨X, F, Y⟩) = (ι h \in D \| (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\})))$
22	21	eqeq1d	$⊢ φ \to (I (⟨X, F, Y⟩) = G \leftrightarrow (ι h \in D \| (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}))) = G)$
23	3 4 5 6 7 8 9 10 11 12 13 14 17 18 15 20 16	mapdpg	$⊢ φ \to \exists! h \in D (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}))$
24		nfv	$⊢ Ⅎ h φ$
25		nfcvd	$⊢ φ \to \underline{Ⅎ} h G$
26		nfvd	$⊢ φ \to Ⅎ h (M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\}))$
27		sneq	$⊢ h = G \to \{h\} = \{G\}$
28	27	fveq2d	$⊢ h = G \to J (\{h\}) = J (\{G\})$
29	28	eqeq2d	$⊢ h = G \to (M (N (\{Y\})) = J (\{h\}) \leftrightarrow M (N (\{Y\})) = J (\{G\}))$
30		oveq2	$⊢ h = G \to F R h = F R G$
31	30	sneqd	$⊢ h = G \to \{F R h\} = \{F R G\}$
32	31	fveq2d	$⊢ h = G \to J (\{F R h\}) = J (\{F R G\})$
33	32	eqeq2d	$⊢ h = G \to (M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}) \leftrightarrow M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\}))$
34	29 33	anbi12d	$⊢ h = G \to ((M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\})) \leftrightarrow (M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})))$
35	34	adantl	$⊢ (φ \land h = G) \to ((M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\})) \leftrightarrow (M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})))$
36	24 25 26 19 35	riota2df	$⊢ (φ \land \exists! h \in D (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}))) \to ((M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})) \leftrightarrow (ι h \in D \| (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}))) = G)$
37	23 36	mpdan	$⊢ φ \to ((M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})) \leftrightarrow (ι h \in D \| (M (N (\{Y\})) = J (\{h\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R h\}))) = G)$
38	22 37	bitr4d	$⊢ φ \to (I (⟨X, F, Y⟩) = G \leftrightarrow (M (N (\{Y\})) = J (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = J (\{F R G\})))$

Metamath Proof Explorer

Theorem mapdheq

Proof