Theorem lclkrlem2h

Description: Lemma for lclkr . Eliminate the ( L( E .+ G ) ) e. J hypothesis. (Contributed by NM, 16-Jan-2015)

	Ref	Expression
Hypotheses	lclkrlem2f.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lclkrlem2f.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
	lclkrlem2f.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lclkrlem2f.v	$$⊢V={\mathrm{Base}}_U$$
	lclkrlem2f.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
	lclkrlem2f.q	$$⊢Q=0_S$$
	lclkrlem2f.z	$$⊢\underset{\dot{}}{0}=0_U$$
	lclkrlem2f.a	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
	lclkrlem2f.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	lclkrlem2f.f	$$⊢F=\mathrm{LFnl}\left(U\right)$$
	lclkrlem2f.j	$$⊢J=\mathrm{LSHyp}\left(U\right)$$
	lclkrlem2f.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
	lclkrlem2f.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
	lclkrlem2f.p	$$⊢\underset{\dot{}}{+}={+}_D$$
	lclkrlem2f.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	lclkrlem2f.b	$$⊢\phi \to B\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lclkrlem2f.e	$$⊢\phi \to E\in F$$
	lclkrlem2f.g	$$⊢\phi \to G\in F$$
	lclkrlem2f.le	$$⊢\phi \to L\left(E\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
	lclkrlem2f.lg	$$⊢\phi \to L\left(G\right)=\underset{\dot{}}{\perp }\left(\left\{Y\right\}\right)$$
	lclkrlem2f.kb	$$⊢\phi \to \left(E\underset{\dot{}}{+}G\right)\left(B\right)=Q$$
	lclkrlem2f.nx	$$⊢\phi \to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\right)$$
	lclkrlem2h.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lclkrlem2h.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lclkrlem2h.ne	$$⊢\phi \to L\left(E\right)\ne L\left(G\right)$$
Assertion	lclkrlem2h	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$

Proof

Step	Hyp	Ref	Expression
1		lclkrlem2f.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lclkrlem2f.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
3		lclkrlem2f.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		lclkrlem2f.v	$$⊢V={\mathrm{Base}}_U$$
5		lclkrlem2f.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
6		lclkrlem2f.q	$$⊢Q=0_S$$
7		lclkrlem2f.z	$$⊢\underset{\dot{}}{0}=0_U$$
8		lclkrlem2f.a	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
9		lclkrlem2f.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
10		lclkrlem2f.f	$$⊢F=\mathrm{LFnl}\left(U\right)$$
11		lclkrlem2f.j	$$⊢J=\mathrm{LSHyp}\left(U\right)$$
12		lclkrlem2f.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
13		lclkrlem2f.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
14		lclkrlem2f.p	$$⊢\underset{\dot{}}{+}={+}_D$$
15		lclkrlem2f.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
16		lclkrlem2f.b	$$⊢\phi \to B\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
17		lclkrlem2f.e	$$⊢\phi \to E\in F$$
18		lclkrlem2f.g	$$⊢\phi \to G\in F$$
19		lclkrlem2f.le	$$⊢\phi \to L\left(E\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
20		lclkrlem2f.lg	$$⊢\phi \to L\left(G\right)=\underset{\dot{}}{\perp }\left(\left\{Y\right\}\right)$$
21		lclkrlem2f.kb	$$⊢\phi \to \left(E\underset{\dot{}}{+}G\right)\left(B\right)=Q$$
22		lclkrlem2f.nx	$$⊢\phi \to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\right)$$
23		lclkrlem2h.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
24		lclkrlem2h.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
25		lclkrlem2h.ne	$$⊢\phi \to L\left(E\right)\ne L\left(G\right)$$
26	15	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
27	16	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to B\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
28	17	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to E\in F$$
29	18	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to G\in F$$
30	19	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to L\left(E\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
31	20	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to L\left(G\right)=\underset{\dot{}}{\perp }\left(\left\{Y\right\}\right)$$
32	21	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \left(E\underset{\dot{}}{+}G\right)\left(B\right)=Q$$
33	22	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\right)$$
34	23	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
35	24	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
36	25	adantr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to L\left(E\right)\ne L\left(G\right)$$
37		simpr	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to L\left(E\underset{\dot{}}{+}G\right)\in J$$
38	1 2 3 4 5 6 7 8 9 10 11 12 13 14 26 27 28 29 30 31 32 33 34 35 36 37	lclkrlem2g	$$⊢\left(\phi \wedge L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$
39	1 3 2 4 15	dochoc1	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(V\right)\right)=V$$
40	39	adantr	$$⊢\left(\phi \wedge \neg L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(V\right)\right)=V$$
41	1 3 15	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
42	1 3 15	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
43	10 13 14 42 17 18	ldualvaddcl	$$⊢\phi \to E\underset{\dot{}}{+}G\in F$$
44	4 11 10 12 41 43	lkrshpor	$$⊢\phi \to \left(L\left(E\underset{\dot{}}{+}G\right)\in J\vee L\left(E\underset{\dot{}}{+}G\right)=V\right)$$
45	44	orcanai	$$⊢\left(\phi \wedge \neg L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to L\left(E\underset{\dot{}}{+}G\right)=V$$
46	45	fveq2d	$$⊢\left(\phi \wedge \neg L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)=\underset{\dot{}}{\perp }\left(V\right)$$
47	46	fveq2d	$$⊢\left(\phi \wedge \neg L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(V\right)\right)$$
48	40 47 45	3eqtr4d	$$⊢\left(\phi \wedge \neg L\left(E\underset{\dot{}}{+}G\right)\in J\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$
49	38 48	pm2.61dan	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$