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Theorem lcfl7N 39515
Description: Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that (𝐿𝐺) = 𝑉 means the functional is zero by lkr0f 37108. (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 + = (+g𝑈)
lcfl6.t · = ( ·𝑠𝑈)
lcfl6.s 𝑆 = (Scalar‘𝑈)
lcfl6.r 𝑅 = (Base‘𝑆)
lcfl6.z 0 = (0g𝑈)
lcfl6.f 𝐹 = (LFnl‘𝑈)
lcfl6.l 𝐿 = (LKer‘𝑈)
lcfl6.c 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
lcfl6.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
lcfl6.g (𝜑𝐺𝐹)
Assertion
Ref Expression
lcfl7N (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{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.
StepHypRef 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 + = (+g𝑈)
6 lcfl6.t . . 3 · = ( ·𝑠𝑈)
7 lcfl6.s . . 3 𝑆 = (Scalar‘𝑈)
8 lcfl6.r . . 3 𝑅 = (Base‘𝑆)
9 lcfl6.z . . 3 0 = (0g𝑈)
10 lcfl6.f . . 3 𝐹 = (LFnl‘𝑈)
11 lcfl6.l . . 3 𝐿 = (LKer‘𝑈)
12 lcfl6.c . . 3 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
13 lcfl6.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
14 lcfl6.g . . 3 (𝜑𝐺𝐹)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfl6 39514 . 2 (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
1613ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
17 eqid 2738 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18 eqid 2738 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19 simplrl 774 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑥 ∈ (𝑉 ∖ { 0 }))
20 simplrr 775 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑦 ∈ (𝑉 ∖ { 0 }))
21 simprl 768 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22 eqeq1 2742 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
2322rexbidv 3226 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
2423riotabidv 7234 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25 oveq1 7282 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
2625oveq2d 7291 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
2726eqeq2d 2749 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
2827rexbidv 3226 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29 oveq1 7282 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
3029eqeq2d 2749 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
3130cbvrexvw 3384 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
3228, 31bitrdi 287 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3332cbvriotavw 7242 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
3424, 33eqtrdi 2794 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3534cbvmptv 5187 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3621, 35eqtrdi 2794 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))))
37 simprr 770 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
38 eqeq1 2742 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
3938rexbidv 3226 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
4039riotabidv 7234 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41 oveq1 7282 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
4241oveq2d 7291 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
4342eqeq2d 2749 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
4443rexbidv 3226 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45 oveq1 7282 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
4645eqeq2d 2749 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
4746cbvrexvw 3384 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
4844, 47bitrdi 287 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
4948cbvriotavw 7242 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
5040, 49eqtrdi 2794 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5150cbvmptv 5187 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5237, 51eqtrdi 2794 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝐺 = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))))
5336, 52eqtr3d 2780 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))))
541, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 53lcfl7lem 39513 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))) → 𝑥 = 𝑦)
5554ex 413 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∖ { 0 }) ∧ 𝑦 ∈ (𝑉 ∖ { 0 }))) → ((𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))
5655ralrimivva 3123 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))
5756a1d 25 . . . . . 6 (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦)))
5857ancld 551 . . . . 5 (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦))))
59 sneq 4571 . . . . . . . . . . 11 (𝑥 = 𝑦 → {𝑥} = {𝑦})
6059fveq2d 6778 . . . . . . . . . 10 (𝑥 = 𝑦 → ( ‘{𝑥}) = ( ‘{𝑦}))
61 oveq2 7283 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
6261oveq2d 7291 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
6362eqeq2d 2749 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
6460, 63rexeqbidv 3337 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6564riotabidv 7234 . . . . . . . 8 (𝑥 = 𝑦 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6665mpteq2dv 5176 . . . . . . 7 (𝑥 = 𝑦 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
6766eqeq2d 2749 . . . . . 6 (𝑥 = 𝑦 → (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
6867reu4 3666 . . . . 5 (∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ ∀𝑥 ∈ (𝑉 ∖ { 0 })∀𝑦 ∈ (𝑉 ∖ { 0 })((𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∧ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))) → 𝑥 = 𝑦)))
6958, 68syl6ibr 251 . . . 4 (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))))
70 reurex 3362 . . . 4 (∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) → ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
7169, 70impbid1 224 . . 3 (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))))
7271orbi2d 913 . 2 (𝜑 → (((𝐿𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) ↔ ((𝐿𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
7315, 72bitrd 278 1 (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  wral 3064  wrex 3065  ∃!wreu 3066  {crab 3068  cdif 3884  {csn 4561  cmpt 5157  cfv 6433  crio 7231  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  Scalarcsca 16965   ·𝑠 cvsca 16966  0gc0g 17150  LFnlclfn 37071  LKerclk 37099  HLchlt 37364  LHypclh 37998  DVecHcdvh 39092  ocHcoch 39361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-riotaBAD 36967
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-undef 8089  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-lmod 20125  df-lss 20194  df-lsp 20234  df-lvec 20365  df-lsatoms 36990  df-lshyp 36991  df-lfl 37072  df-lkr 37100  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-tgrp 38757  df-tendo 38769  df-edring 38771  df-dveca 39017  df-disoa 39043  df-dvech 39093  df-dib 39153  df-dic 39187  df-dih 39243  df-doch 39362  df-djh 39409
This theorem is referenced by: (None)
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