Theorem lclkrlem2i

Description: Lemma for lclkr . Eliminate the ( LE ) =/= ( LG ) hypothesis. (Contributed by NM, 17-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	$$⊢\phi \to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\right)$$
	lclkrlem2i.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lclkrlem2i.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
Assertion	lclkrlem2i	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$

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		lclkrlem2i.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
24		lclkrlem2i.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
25	15	adantr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
26	23	adantr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
27	17	adantr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to E\in F$$
28	18	adantr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to G\in F$$
29	19	adantr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to L\left(E\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
30		simpr	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to L\left(E\right)=L\left(G\right)$$
31	1 2 3 4 7 10 12 13 14 25 26 27 28 29 30	lclkrlem2e	$$⊢\left(\phi \wedge L\left(E\right)=L\left(G\right)\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$
32	15	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
33	16	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to B\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
34	17	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to E\in F$$
35	18	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to G\in F$$
36	19	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to L\left(E\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
37	20	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to L\left(G\right)=\underset{\dot{}}{\perp }\left(\left\{Y\right\}\right)$$
38	21	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to \left(E\underset{\dot{}}{+}G\right)\left(B\right)=Q$$
39	22	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{B\right\}\right)\right)$$
40	23	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
41	24	adantr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
42		simpr	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to L\left(E\right)\ne L\left(G\right)$$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 33 34 35 36 37 38 39 40 41 42	lclkrlem2h	$$⊢\left(\phi \wedge L\left(E\right)\ne L\left(G\right)\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$
44	31 43	pm2.61dane	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(E\underset{\dot{}}{+}G\right)\right)\right)=L\left(E\underset{\dot{}}{+}G\right)$$