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Theorem lcfl7N 41484
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 39076. (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 41483 . 2 (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
1613ad2antrr 726 . . . . . . . . . 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 2735 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18 eqid 2735 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19 simplrl 777 . . . . . . . . . 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 778 . . . . . . . . . 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 771 . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
2322rexbidv 3177 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
2423riotabidv 7390 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25 oveq1 7438 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
2625oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
2726eqeq2d 2746 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
2827rexbidv 3177 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
3029eqeq2d 2746 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
3130cbvrexvw 3236 . . . . . . . . . . . . . . . 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 7398 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
3424, 33eqtrdi 2791 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3534cbvmptv 5261 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3621, 35eqtrdi 2791 . . . . . . . . . . 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 773 . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
3938rexbidv 3177 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
4039riotabidv 7390 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41 oveq1 7438 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
4241oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
4342eqeq2d 2746 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
4443rexbidv 3177 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
4645eqeq2d 2746 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
4746cbvrexvw 3236 . . . . . . . . . . . . . . . 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 7398 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
5040, 49eqtrdi 2791 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5150cbvmptv 5261 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5237, 51eqtrdi 2791 . . . . . . . . . . 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 2777 . . . . . . . . . 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 41482 . . . . . . . . 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 412 . . . . . . . 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 3200 . . . . . . 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 550 . . . . 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 4641 . . . . . . . . . . 11 (𝑥 = 𝑦 → {𝑥} = {𝑦})
6059fveq2d 6911 . . . . . . . . . 10 (𝑥 = 𝑦 → ( ‘{𝑥}) = ( ‘{𝑦}))
61 oveq2 7439 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
6261oveq2d 7447 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
6362eqeq2d 2746 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
6460, 63rexeqbidv 3345 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6564riotabidv 7390 . . . . . . . 8 (𝑥 = 𝑦 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6665mpteq2dv 5250 . . . . . . 7 (𝑥 = 𝑦 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
6766eqeq2d 2746 . . . . . 6 (𝑥 = 𝑦 → (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
6867reu4 3740 . . . . 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, 68imbitrrdi 252 . . . 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 3382 . . . 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 225 . . 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 915 . 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 279 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376  {crab 3433  cdif 3960  {csn 4631  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  LFnlclfn 39039  LKerclk 39067  HLchlt 39332  LHypclh 39967  DVecHcdvh 41061  ocHcoch 41330
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-undef 8297  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-0g 17488  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lvec 21120  df-lsatoms 38958  df-lshyp 38959  df-lfl 39040  df-lkr 39068  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tgrp 40726  df-tendo 40738  df-edring 40740  df-dveca 40986  df-disoa 41012  df-dvech 41062  df-dib 41122  df-dic 41156  df-dih 41212  df-doch 41331  df-djh 41378
This theorem is referenced by: (None)
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