Theorem lcfl7N 39964

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 37556. (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 39963	. 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 2736	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18		eqid 2736	. . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
23	22	rexbidv 3175	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
24	23	riotabidv 7315	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
26	25	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
27	26	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
28	27	rexbidv 3175	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29		oveq1 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
30	29	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
31	30	cbvrexvw 3226	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
32	28, 31	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
33	32	cbvriotavw 7323	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
34	24, 33	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
35	34	cbvmptv 5218	. . . . . . . . . . . 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 2792	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
39	38	rexbidv 3175	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
40	39	riotabidv 7315	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41		oveq1 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
42	41	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
43	42	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
44	43	rexbidv 3175	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45		oveq1 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
46	45	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
47	46	cbvrexvw 3226	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
48	44, 47	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
49	48	cbvriotavw 7323	. . . . . . . . . . . . . 14 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
50	40, 49	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
51	50	cbvmptv 5218	. . . . . . . . . . . 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 2792	. . . . . . . . . . 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 2778	. . . . . . . . . 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 39962	. . . . . . . . 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 413	. . . . . . . 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 3197	. . . . . . 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 551	. . . . 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 4596	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
60	59	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑦}))
61		oveq2 7365	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
62	61	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
63	62	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
64	60, 63	rexeqbidv 3320	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
65	64	riotabidv 7315	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
66	65	mpteq2dv 5207	. . . . . . 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 2747	. . . . . 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 3689	. . . . 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 251	. . . 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 3357	. . . 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 224	. . 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 914	. 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 278	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 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  ∃!wreu 3351  {crab 3407   ∖ cdif 3907  {csn 4586   ↦ cmpt 5188  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321  LFnlclfn 37519  LKerclk 37547  HLchlt 37812  LHypclh 38447  DVecHcdvh 39541  ocHcoch 39810

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-lsatoms 37438  df-lshyp 37439  df-lfl 37520  df-lkr 37548  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tgrp 39206  df-tendo 39218  df-edring 39220  df-dveca 39466  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692  df-doch 39811  df-djh 39858

This theorem is referenced by: (None)