Theorem lcfl7N 41866

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 39459. (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 41865	. 2 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
16	13	ad2antrr 727	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18		eqid 2737	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19		simplrl 777	. . . . . . . . . 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 778	. . . . . . . . . 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 771	. . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
23	22	rexbidv 3162	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
24	23	riotabidv 7327	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25		oveq1 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
26	25	oveq2d 7384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
27	26	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
28	27	rexbidv 3162	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29		oveq1 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
30	29	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
31	30	cbvrexvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
32	28, 31	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
33	32	cbvriotavw 7335	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
34	24, 33	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
35	34	cbvmptv 5204	. . . . . . . . . . . 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 2788	. . . . . . . . . . 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 773	. . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
39	38	rexbidv 3162	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
40	39	riotabidv 7327	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41		oveq1 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
42	41	oveq2d 7384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
43	42	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
44	43	rexbidv 3162	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45		oveq1 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
46	45	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
47	46	cbvrexvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
48	44, 47	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
49	48	cbvriotavw 7335	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
50	40, 49	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
51	50	cbvmptv 5204	. . . . . . . . . . . 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 2788	. . . . . . . . . . 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 2774	. . . . . . . . . 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 41864	. . . . . . . . 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 412	. . . . . . . 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 3181	. . . . . . 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 25	. . . . . 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 550	. . . . 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 4592	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
60	59	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑦}))
61		oveq2 7376	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
62	61	oveq2d 7384	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
63	62	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
64	60, 63	rexeqbidv 3319	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
65	64	riotabidv 7327	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
66	65	mpteq2dv 5194	. . . . . . 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 2748	. . . . . 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 3691	. . . . 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 252	. . . 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 3356	. . . 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 225	. . 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 916	. 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 279	1 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  {crab 3401   ∖ cdif 3900  {csn 4582   ↦ cmpt 5181  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371  LFnlclfn 39422  LKerclk 39450  HLchlt 39715  LHypclh 40349  DVecHcdvh 41443  ocHcoch 41712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-riotaBAD 39318

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-undef 8225  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-0g 17373  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19065  df-cntz 19258  df-lsm 19577  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20285  df-dvdsr 20305  df-unit 20306  df-invr 20336  df-dvr 20349  df-drng 20676  df-lmod 20825  df-lss 20895  df-lsp 20935  df-lvec 21067  df-lsatoms 39341  df-lshyp 39342  df-lfl 39423  df-lkr 39451  df-oposet 39541  df-ol 39543  df-oml 39544  df-covers 39631  df-ats 39632  df-atl 39663  df-cvlat 39687  df-hlat 39716  df-llines 39863  df-lplanes 39864  df-lvols 39865  df-lines 39866  df-psubsp 39868  df-pmap 39869  df-padd 40161  df-lhyp 40353  df-laut 40354  df-ldil 40469  df-ltrn 40470  df-trl 40524  df-tgrp 41108  df-tendo 41120  df-edring 41122  df-dveca 41368  df-disoa 41394  df-dvech 41444  df-dib 41504  df-dic 41538  df-dih 41594  df-doch 41713  df-djh 41760

This theorem is referenced by: (None)