Theorem mapdpglem31

Description: Lemma for mapdpg . Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015)

	Ref	Expression
Hypotheses	mapdpg.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdpg.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdpg.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdpg.v	$$⊢V={\mathrm{Base}}_U$$
	mapdpg.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdpg.z	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdpg.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdpg.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdpg.f	$$⊢F={\mathrm{Base}}_C$$
	mapdpg.r	$$⊢R={-}_C$$
	mapdpg.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdpg.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdpg.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdpg.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdpg.g	$$⊢\phi \to G\in F$$
	mapdpg.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
	mapdpg.e	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
	mapdpgem25.h1	$$⊢\phi \to \left(h\in F\wedge \left(M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{GRh\right\}\right)\right)\right)$$
	mapdpgem25.i1	$$⊢\phi \to \left(i\in F\wedge \left(M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{i\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{GRi\right\}\right)\right)\right)$$
	mapdpglem26.a	$$⊢A=\mathrm{Scalar}\left(U\right)$$
	mapdpglem26.b	$$⊢B={\mathrm{Base}}_A$$
	mapdpglem26.t	$$⊢\underset{\dot{}}{·}={\cdot }_C$$
	mapdpglem26.o	$$⊢O=0_A$$
	mapdpglem28.ve	$$⊢\phi \to v\in B$$
	mapdpglem28.u1	$$⊢\phi \to h=u\underset{\dot{}}{·}i$$
	mapdpglem28.u2	$$⊢\phi \to GRh=v\underset{\dot{}}{·}\left(GRi\right)$$
	mapdpglem28.ue	$$⊢\phi \to u\in B$$
Assertion	mapdpglem31	$$⊢\phi \to h=i$$

Proof

Step	Hyp	Ref	Expression
1		mapdpg.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		mapdpg.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
3		mapdpg.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		mapdpg.v	$$⊢V={\mathrm{Base}}_U$$
5		mapdpg.s	$$⊢\underset{\dot{}}{-}={-}_U$$
6		mapdpg.z	$$⊢\underset{\dot{}}{0}=0_U$$
7		mapdpg.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
8		mapdpg.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
9		mapdpg.f	$$⊢F={\mathrm{Base}}_C$$
10		mapdpg.r	$$⊢R={-}_C$$
11		mapdpg.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
12		mapdpg.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
13		mapdpg.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
14		mapdpg.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
15		mapdpg.g	$$⊢\phi \to G\in F$$
16		mapdpg.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
17		mapdpg.e	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
18		mapdpgem25.h1	$$⊢\phi \to \left(h\in F\wedge \left(M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{GRh\right\}\right)\right)\right)$$
19		mapdpgem25.i1	$$⊢\phi \to \left(i\in F\wedge \left(M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{i\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{GRi\right\}\right)\right)\right)$$
20		mapdpglem26.a	$$⊢A=\mathrm{Scalar}\left(U\right)$$
21		mapdpglem26.b	$$⊢B={\mathrm{Base}}_A$$
22		mapdpglem26.t	$$⊢\underset{\dot{}}{·}={\cdot }_C$$
23		mapdpglem26.o	$$⊢O=0_A$$
24		mapdpglem28.ve	$$⊢\phi \to v\in B$$
25		mapdpglem28.u1	$$⊢\phi \to h=u\underset{\dot{}}{·}i$$
26		mapdpglem28.u2	$$⊢\phi \to GRh=v\underset{\dot{}}{·}\left(GRi\right)$$
27		mapdpglem28.ue	$$⊢\phi \to u\in B$$
28		eqid	$$⊢1_A=1_A$$
29		eqid	$$⊢\mathrm{Scalar}\left(C\right)=\mathrm{Scalar}\left(C\right)$$
30		eqid	$$⊢1_{\mathrm{Scalar}\left(C\right)}=1_{\mathrm{Scalar}\left(C\right)}$$
31	1 3 20 28 8 29 30 12	lcd1	$$⊢\phi \to 1_{\mathrm{Scalar}\left(C\right)}=1_A$$
32	31	oveq1d	$$⊢\phi \to 1_{\mathrm{Scalar}\left(C\right)}\underset{\dot{}}{·}i=1_A\underset{\dot{}}{·}i$$
33	1 8 12	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
34	19	simpld	$$⊢\phi \to i\in F$$
35	9 29 22 30	lmodvs1	$$⊢\left(C\in \mathrm{LMod}\wedge i\in F\right)\to 1_{\mathrm{Scalar}\left(C\right)}\underset{\dot{}}{·}i=i$$
36	33 34 35	syl2anc	$$⊢\phi \to 1_{\mathrm{Scalar}\left(C\right)}\underset{\dot{}}{·}i=i$$
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27	mapdpglem30	$$⊢\phi \to \left(v=1_A\wedge v=u\right)$$
38		eqtr2	$$⊢\left(v=1_A\wedge v=u\right)\to 1_A=u$$
39	37 38	syl	$$⊢\phi \to 1_A=u$$
40	39	oveq1d	$$⊢\phi \to 1_A\underset{\dot{}}{·}i=u\underset{\dot{}}{·}i$$
41	32 36 40	3eqtr3rd	$$⊢\phi \to u\underset{\dot{}}{·}i=i$$
42	25 41	eqtrd	$$⊢\phi \to h=i$$