Theorem lcfl7N 42750

Description: Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that (𝐿‘𝐺) = 𝑉 means the functional is zero by lkr0f 40343. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lcfl6.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfl6.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfl6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfl6.v	⊢ 𝑉 = (Base‘𝑈)
lcfl6.a	⊢ + = (+g‘𝑈)
lcfl6.t	⊢ · = ( ·𝑠 ‘𝑈)
lcfl6.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfl6.r	⊢ 𝑅 = (Base‘𝑆)
lcfl6.z	⊢ 0 = (0g‘𝑈)
lcfl6.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfl6.l	⊢ 𝐿 = (LKer‘𝑈)
lcfl6.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcfl6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfl6.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Assertion

Ref	Expression
lcfl7N	⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))

Distinct variable groups:   𝑣,𝑘,𝑤, +   𝑓,𝑘,𝑣,𝑤,𝑥, ⊥   𝑤, 0 ,𝑥   𝑥,𝐶   𝑓,𝐺,𝑥   𝑓,𝐹   𝑓,𝐿,𝑥   𝜑,𝑥   𝑅,𝑘,𝑣   𝑆,𝑘,𝑤,𝑥   𝑣,𝑉,𝑥   𝑥,𝑈   · ,𝑘,𝑣,𝑤   𝑥, +   𝑥,𝑅   𝑥, ·

Allowed substitution hints:   𝜑(𝑤, 𝑣, 𝑓, 𝑘)   𝐶(𝑤, 𝑣, 𝑓, 𝑘)   + (𝑓)   𝑅(𝑤, 𝑓)   𝑆(𝑣, 𝑓)   · (𝑓)   𝑈(𝑤, 𝑣, 𝑓, 𝑘)   𝐹(𝑥, 𝑤, 𝑣, 𝑘)   𝐺(𝑤, 𝑣, 𝑘)   𝐻(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐾(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐿(𝑤, 𝑣, 𝑘)   𝑉(𝑤, 𝑓, 𝑘)   𝑊(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   0 (𝑣, 𝑓, 𝑘)

Proof of Theorem lcfl7N

Dummy variables 𝑙 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfl6.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfl6.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfl6.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfl6.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lcfl6.a	. . 3 ⊢ + = (+g‘𝑈)
6		lcfl6.t	. . 3 ⊢ · = ( ·𝑠 ‘𝑈)
7		lcfl6.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
8		lcfl6.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
9		lcfl6.z	. . 3 ⊢ 0 = (0g‘𝑈)
10		lcfl6.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
11		lcfl6.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
12		lcfl6.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
13		lcfl6.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		lcfl6.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfl6 42749	. 2 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
16	13	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		eqid 2763	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18		eqid 2763	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑥 ∈ (𝑉 ∖ { 0 }))
20		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑦 ∈ (𝑉 ∖ { 0 }))
21		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22		eqeq1 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
23	22	rexbidv 3190	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
24	23	riotabidv 7402	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25		oveq1 7450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
26	25	oveq2d 7459	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
27	26	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
28	27	rexbidv 3190	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29		oveq1 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
30	29	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
31	30	cbvrexvw 3245	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
32	28, 31	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
33	32	cbvriotavw 7410	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
34	24, 33	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
35	34	cbvmptv 5212	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
36	21, 35	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))))
37		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
38		eqeq1 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
39	38	rexbidv 3190	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
40	39	riotabidv 7402	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41		oveq1 7450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
42	41	oveq2d 7459	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
43	42	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
44	43	rexbidv 3190	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45		oveq1 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
46	45	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
47	46	cbvrexvw 3245	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
48	44, 47	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
49	48	cbvriotavw 7410	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
50	40, 49	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
51	50	cbvmptv 5212	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
52	37, 51	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))))
53	36, 52	eqtr3d 2800	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))))
54	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 53	lcfl7lem 42748	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑥 = 𝑦)
55	54	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) → ((𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))
56	55	ralrimivva 3209	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))
57	56	a1d 26	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦)))
58	57	ancld 560	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))))
59		sneq 4594	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
60	59	fveq2d 6912	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑦}))
61		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
62	61	oveq2d 7459	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
63	62	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
64	60, 63	rexeqbidv 3339	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
65	64	riotabidv 7402	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
66	65	mpteq2dv 5202	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
67	66	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
68	67	reu4 3692	. . . . 5 ⊢ (∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦)))
69	58, 68	imbitrrdi 255	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))))
70		reurex 3373	. . . 4 ⊢ (∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
71	69, 70	impbid1 228	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))))
72	71	orbi2d 929	. 2 ⊢ (𝜑 → (((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
73	15, 72	bitrd 282	1 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367  {crab 3416   ∖ cdif 4048  {csn 4588   ↦ cmpt 5189  ‘cfv 6558  ℩crio 7399  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  Scalarcsca 17460   ·_𝑠 cvsca 17461  0_gc0g 17639  LFnlclfn 40306  LKerclk 40334  HLchlt 40599  LHypclh 41233  DVecHcdvh 42327  ocHcoch 42596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-riotaBAD 40202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-tpos 8271  df-undef 8318  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-n0 12634  df-z 12721  df-uz 12993  df-fz 13667  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-0g 17641  df-proset 18497  df-poset 18516  df-plt 18531  df-lub 18547  df-glb 18548  df-join 18549  df-meet 18550  df-p0 18626  df-p1 18627  df-lat 18635  df-clat 18702  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-subg 19362  df-cntz 19561  df-lsm 19880  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-oppr 20597  df-dvdsr 20617  df-unit 20618  df-invr 20648  df-dvr 20661  df-drng 21012  df-lmod 21167  df-lss 21237  df-lsp 21277  df-lvec 21408  df-lsatoms 40225  df-lshyp 40226  df-lfl 40307  df-lkr 40335  df-oposet 40425  df-ol 40427  df-oml 40428  df-covers 40515  df-ats 40516  df-atl 40547  df-cvlat 40571  df-hlat 40600  df-llines 40747  df-lplanes 40748  df-lvols 40749  df-lines 40750  df-psubsp 40752  df-pmap 40753  df-padd 41045  df-lhyp 41237  df-laut 41238  df-ldil 41353  df-ltrn 41354  df-trl 41408  df-tgrp 41992  df-tendo 42004  df-edring 42006  df-dveca 42252  df-disoa 42278  df-dvech 42328  df-dib 42388  df-dic 42422  df-dih 42478  df-doch 42597  df-djh 42644

This theorem is used by: (None)