Theorem lcfl7N 38631

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 36224. (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 38630	. 2 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
16	13	ad2antrr 724	. . . . . . . . . 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 2821	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18		eqid 2821	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19		simplrl 775	. . . . . . . . . 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 776	. . . . . . . . . 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 769	. . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
23	22	rexbidv 3297	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
24	23	riotabidv 7110	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25		oveq1 7157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
26	25	oveq2d 7166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
27	26	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
28	27	rexbidv 3297	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29		oveq1 7157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
30	29	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
31	30	cbvrexvw 3451	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
32	28, 31	syl6bb 289	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
33	32	cbvriotavw 7118	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
34	24, 33	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
35	34	cbvmptv 5162	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
36	21, 35	syl6eq 2872	. . . . . . . . . . 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 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 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
38		eqeq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
39	38	rexbidv 3297	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
40	39	riotabidv 7110	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41		oveq1 7157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
42	41	oveq2d 7166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
43	42	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
44	43	rexbidv 3297	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45		oveq1 7157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
46	45	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
47	46	cbvrexvw 3451	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
48	44, 47	syl6bb 289	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
49	48	cbvriotavw 7118	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
50	40, 49	syl6eq 2872	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
51	50	cbvmptv 5162	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
52	37, 51	syl6eq 2872	. . . . . . . . . . 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 2858	. . . . . . . . . 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 38629	. . . . . . . . 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 415	. . . . . . . 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 3191	. . . . . . 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 553	. . . . 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 4571	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
60	59	fveq2d 6669	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑦}))
61		oveq2 7158	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
62	61	oveq2d 7166	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
63	62	eqeq2d 2832	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
64	60, 63	rexeqbidv 3403	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
65	64	riotabidv 7110	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
66	65	mpteq2dv 5155	. . . . . . 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 2832	. . . . . 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 3722	. . . . 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	syl6ibr 254	. . . 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 3432	. . . 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 227	. . 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 912	. 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 281	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 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142   ∖ cdif 3933  {csn 4561   ↦ cmpt 5139  ‘cfv 6350  ℩crio 7107  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  Scalarcsca 16562   ·_𝑠 cvsca 16563  0_gc0g 16707  LFnlclfn 36187  LKerclk 36215  HLchlt 36480  LHypclh 37114  DVecHcdvh 38208  ocHcoch 38477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-riotaBAD 36083

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-tpos 7886  df-undef 7933  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-0g 16709  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-p1 17644  df-lat 17650  df-clat 17712  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-grp 18100  df-minusg 18101  df-sbg 18102  df-subg 18270  df-cntz 18441  df-lsm 18755  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-ring 19293  df-oppr 19367  df-dvdsr 19385  df-unit 19386  df-invr 19416  df-dvr 19427  df-drng 19498  df-lmod 19630  df-lss 19698  df-lsp 19738  df-lvec 19869  df-lsatoms 36106  df-lshyp 36107  df-lfl 36188  df-lkr 36216  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-llines 36628  df-lplanes 36629  df-lvols 36630  df-lines 36631  df-psubsp 36633  df-pmap 36634  df-padd 36926  df-lhyp 37118  df-laut 37119  df-ldil 37234  df-ltrn 37235  df-trl 37289  df-tgrp 37873  df-tendo 37885  df-edring 37887  df-dveca 38133  df-disoa 38159  df-dvech 38209  df-dib 38269  df-dic 38303  df-dih 38359  df-doch 38478  df-djh 38525

This theorem is referenced by: (None)