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Theorem lcfl7N 37309
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 34901. (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 37308 . 2 (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
1613ad2antrr 705 . . . . . . . . . 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 2771 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
18 eqid 2771 . . . . . . . . . 10 (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
19 simplrl 762 . . . . . . . . . 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 763 . . . . . . . . . 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 754 . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑥))))
2322rexbidv 3200 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
2423riotabidv 6759 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))))
25 oveq1 6803 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥))
2625oveq2d 6812 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑙 · 𝑥)))
2726eqeq2d 2781 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑥))))
2827rexbidv 3200 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥))))
29 oveq1 6803 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑥)))
3029eqeq2d 2781 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥))))
3130cbvrexv 3321 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
3228, 31syl6bb 276 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3332cbvriotav 6768 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑢 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))
3424, 33syl6eq 2821 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3534cbvmptv 4885 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))))
3621, 35syl6eq 2821 . . . . . . . . . . 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 756 . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑘 · 𝑦))))
3938rexbidv 3200 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
4039riotabidv 6759 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))))
41 oveq1 6803 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑙 → (𝑘 · 𝑦) = (𝑙 · 𝑦))
4241oveq2d 6812 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑙 · 𝑦)))
4342eqeq2d 2781 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑢 = (𝑤 + (𝑙 · 𝑦))))
4443rexbidv 3200 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦))))
45 oveq1 6803 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤 + (𝑙 · 𝑦)) = (𝑧 + (𝑙 · 𝑦)))
4645eqeq2d 2781 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦))))
4746cbvrexv 3321 . . . . . . . . . . . . . . . 16 (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑙 · 𝑦)) ↔ ∃𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
4844, 47syl6bb 276 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (∃𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
4948cbvriotav 6768 . . . . . . . . . . . . . 14 (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑢 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))
5040, 49syl6eq 2821 . . . . . . . . . . . . 13 (𝑣 = 𝑢 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5150cbvmptv 4885 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑢𝑉 ↦ (𝑙𝑅𝑧 ∈ ( ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))
5237, 51syl6eq 2821 . . . . . . . . . . 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 2807 . . . . . . . . . 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 37307 . . . . . . . . 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 397 . . . . . . . 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 3120 . . . . . . 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 540 . . . . 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 4327 . . . . . . . . . . 11 (𝑥 = 𝑦 → {𝑥} = {𝑦})
6059fveq2d 6337 . . . . . . . . . 10 (𝑥 = 𝑦 → ( ‘{𝑥}) = ( ‘{𝑦}))
61 oveq2 6804 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑘 · 𝑥) = (𝑘 · 𝑦))
6261oveq2d 6812 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑦)))
6362eqeq2d 2781 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑦))))
6460, 63rexeqbidv 3302 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6564riotabidv 6759 . . . . . . . 8 (𝑥 = 𝑦 → (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))
6665mpteq2dv 4880 . . . . . . 7 (𝑥 = 𝑦 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
6766eqeq2d 2781 . . . . . 6 (𝑥 = 𝑦 → (𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ↔ 𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
6867reu4 3552 . . . . 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 242 . . . 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 3309 . . . 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 215 . . 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 901 . 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 268 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 196  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ∃!wreu 3063  {crab 3065  cdif 3720  {csn 4317  cmpt 4864  cfv 6030  crio 6756  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16308  LFnlclfn 34864  LKerclk 34892  HLchlt 35157  LHypclh 35791  DVecHcdvh 36886  ocHcoch 37155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-riotaBAD 34759
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-tpos 7508  df-undef 7555  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-0g 16310  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-cntz 17957  df-lsm 18258  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-oppr 18831  df-dvdsr 18849  df-unit 18850  df-invr 18880  df-dvr 18891  df-drng 18959  df-lmod 19075  df-lss 19143  df-lsp 19185  df-lvec 19316  df-lsatoms 34783  df-lshyp 34784  df-lfl 34865  df-lkr 34893  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967  df-tgrp 36551  df-tendo 36563  df-edring 36565  df-dveca 36811  df-disoa 36837  df-dvech 36887  df-dib 36947  df-dic 36981  df-dih 37037  df-doch 37156  df-djh 37203
This theorem is referenced by: (None)
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