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Theorem lcfrlem9 37156
Description: Lemma for lcf1o 37157. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 37157.) TODO: ugly proof; maybe have better subtheorems or abbreviate some 𝑘 expansions with 𝐽𝑧? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
Hypotheses
Ref Expression
lcf1o.h 𝐻 = (LHyp‘𝐾)
lcf1o.o = ((ocH‘𝐾)‘𝑊)
lcf1o.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcf1o.v 𝑉 = (Base‘𝑈)
lcf1o.a + = (+g𝑈)
lcf1o.t · = ( ·𝑠𝑈)
lcf1o.s 𝑆 = (Scalar‘𝑈)
lcf1o.r 𝑅 = (Base‘𝑆)
lcf1o.z 0 = (0g𝑈)
lcf1o.f 𝐹 = (LFnl‘𝑈)
lcf1o.l 𝐿 = (LKer‘𝑈)
lcf1o.d 𝐷 = (LDual‘𝑈)
lcf1o.q 𝑄 = (0g𝐷)
lcf1o.c 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
lcf1o.j 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
Assertion
Ref Expression
lcfrlem9 (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Distinct variable groups:   𝑥,𝑤,   𝑥, 0 ,𝑣   𝑣,𝑉,𝑥   𝑥, ·   𝑣,𝑘,𝑤,𝑥, +   𝑥,𝑅   𝑓,𝑘,𝑣,𝑤,𝑥, +   𝑘,𝐽,𝑣,𝑤,𝑥   𝐶,𝑘,𝑣,𝑤,𝑥   𝑓,𝐹   𝑓,𝐿,𝑘,𝑣,𝑤,𝑥   ,𝑓,𝑘,𝑣   𝑄,𝑘,𝑣,𝑤,𝑥   𝑅,𝑓,𝑘,𝑣,𝑤   𝑆,𝑘,𝑣,𝑤,𝑥   · ,𝑓,𝑘,𝑣,𝑤   𝑈,𝑘,𝑤,𝑥   𝑓,𝑉,𝑘,𝑤   0 ,𝑘,𝑣,𝑤   𝜑,𝑘,𝑣,𝑤,𝑥
Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑥,𝑤,𝑣,𝑓,𝑘)   𝑄(𝑓)   𝑆(𝑓)   𝑈(𝑣,𝑓)   𝐹(𝑥,𝑤,𝑣,𝑘)   𝐻(𝑥,𝑤,𝑣,𝑓,𝑘)   𝐽(𝑓)   𝐾(𝑥,𝑤,𝑣,𝑓,𝑘)   𝑊(𝑥,𝑤,𝑣,𝑓,𝑘)   0 (𝑓)

Proof of Theorem lcfrlem9
Dummy variables 𝑦 𝑔 𝑡 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcf1o.v . . . . . 6 𝑉 = (Base‘𝑈)
2 fvex 6239 . . . . . 6 (Base‘𝑈) ∈ V
31, 2eqeltri 2726 . . . . 5 𝑉 ∈ V
43mptex 6527 . . . 4 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∈ V
5 lcf1o.j . . . 4 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
64, 5fnmpti 6060 . . 3 𝐽 Fn (𝑉 ∖ { 0 })
76a1i 11 . 2 (𝜑𝐽 Fn (𝑉 ∖ { 0 }))
8 fvelrnb 6282 . . . . 5 (𝐽 Fn (𝑉 ∖ { 0 }) → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
97, 8syl 17 . . . 4 (𝜑 → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
10 lcf1o.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
11 lcf1o.o . . . . . . . . 9 = ((ocH‘𝐾)‘𝑊)
12 lcf1o.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
13 lcf1o.a . . . . . . . . 9 + = (+g𝑈)
14 lcf1o.t . . . . . . . . 9 · = ( ·𝑠𝑈)
15 lcf1o.s . . . . . . . . 9 𝑆 = (Scalar‘𝑈)
16 lcf1o.r . . . . . . . . 9 𝑅 = (Base‘𝑆)
17 lcf1o.z . . . . . . . . 9 0 = (0g𝑈)
18 lcf1o.f . . . . . . . . 9 𝐹 = (LFnl‘𝑈)
19 lcf1o.l . . . . . . . . 9 𝐿 = (LKer‘𝑈)
20 lcf1o.d . . . . . . . . 9 𝐷 = (LDual‘𝑈)
21 lcf1o.q . . . . . . . . 9 𝑄 = (0g𝐷)
22 lcf1o.c . . . . . . . . 9 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
23 lcflo.k . . . . . . . . . 10 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2423adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
25 simpr 476 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
2610, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 5, 24, 25lcfrlem8 37155 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
27 eqid 2651 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
28 sneq 4220 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2928fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ( ‘{𝑦}) = ( ‘{𝑧}))
30 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
3130oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
3231eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
3329, 32rexeqbidv 3183 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3433riotabidv 6653 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3534mpteq2dv 4778 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
3635eqeq2d 2661 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ((𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
3736rspcev 3340 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
3825, 27, 37sylancl 695 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
3938olcd 407 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ∨ ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
4010, 11, 12, 1, 17, 13, 14, 18, 15, 16, 27, 24, 25dochflcl 37081 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐹)
4110, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 22, 24, 40lcfl6 37106 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶 ↔ ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ∨ ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))))
4239, 41mpbird 247 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶)
4310, 11, 12, 1, 17, 13, 14, 19, 15, 16, 27, 24, 25dochsnkr2cl 37080 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))) ∖ { 0 }))
4410, 11, 12, 1, 17, 18, 19, 24, 40, 43dochsnkrlem3 37077 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) = (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
4510, 11, 12, 1, 17, 18, 19, 24, 40, 43dochsnkrlem1 37075 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
4644, 45eqnetrrd 2891 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
4710, 12, 23dvhlmod 36716 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ LMod)
4847adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
491, 18, 19, 20, 21, 48, 40lkr0f2 34766 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = 𝑄))
5049necon3bid 2867 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
5146, 50mpbid 222 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
52 eldifsn 4350 . . . . . . . . 9 ((𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}) ↔ ((𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶 ∧ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
5342, 51, 52sylanbrc 699 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
5426, 53eqeltrd 2730 . . . . . . 7 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) ∈ (𝐶 ∖ {𝑄}))
55 eleq1 2718 . . . . . . 7 ((𝐽𝑧) = 𝑔 → ((𝐽𝑧) ∈ (𝐶 ∖ {𝑄}) ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
5654, 55syl5ibcom 235 . . . . . 6 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
5756rexlimdva 3060 . . . . 5 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
58 eldifsn 4350 . . . . . . . 8 (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔𝐶𝑔𝑄))
59 simprl 809 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐶)
6047adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑈 ∈ LMod)
6122lcfl1lem 37097 . . . . . . . . . . . . . . . 16 (𝑔𝐶 ↔ (𝑔𝐹 ∧ ( ‘( ‘(𝐿𝑔))) = (𝐿𝑔)))
6261simplbi 475 . . . . . . . . . . . . . . 15 (𝑔𝐶𝑔𝐹)
6362adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔𝐹)
641, 18, 19, 20, 21, 60, 63lkr0f2 34766 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → ((𝐿𝑔) = 𝑉𝑔 = 𝑄))
6564necon3bid 2867 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → ((𝐿𝑔) ≠ 𝑉𝑔𝑄))
6665biimprd 238 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → (𝑔𝑄 → (𝐿𝑔) ≠ 𝑉))
6766impr 648 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐿𝑔) ≠ 𝑉)
6867neneqd 2828 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ¬ (𝐿𝑔) = 𝑉)
6923adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7062adantr 480 . . . . . . . . . . . . . 14 ((𝑔𝐶𝑔𝑄) → 𝑔𝐹)
7170adantl 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐹)
7210, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 22, 69, 71lcfl6 37106 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝑔𝐶 ↔ ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))))
7372biimpa 500 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
7473ord 391 . . . . . . . . . 10 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → (¬ (𝐿𝑔) = 𝑉 → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
75743impia 1280 . . . . . . . . 9 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶 ∧ ¬ (𝐿𝑔) = 𝑉) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7659, 68, 75mpd3an23 1466 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7758, 76sylan2b 491 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
78 eqcom 2658 . . . . . . . . 9 ((𝐽𝑧) = 𝑔𝑔 = (𝐽𝑧))
7923ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
80 simpr 476 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
8110, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 5, 79, 80lcfrlem8 37155 . . . . . . . . . 10 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
8281eqeq2d 2661 . . . . . . . . 9 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑔 = (𝐽𝑧) ↔ 𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8378, 82syl5bb 272 . . . . . . . 8 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8483rexbidva 3078 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8577, 84mpbird 247 . . . . . 6 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔)
8685ex 449 . . . . 5 (𝜑 → (𝑔 ∈ (𝐶 ∖ {𝑄}) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
8757, 86impbid 202 . . . 4 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
889, 87bitrd 268 . . 3 (𝜑 → (𝑔 ∈ ran 𝐽𝑔 ∈ (𝐶 ∖ {𝑄})))
8988eqrdv 2649 . 2 (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
9023ad2antrr 762 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
91 eqid 2651 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
92 eqid 2651 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
93 simplrl 817 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
94 simplrr 818 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
95 simpr 476 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑡) = (𝐽𝑢))
9610, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 5, 90, 93lcfrlem8 37155 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑡) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))))
9710, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 5, 90, 94lcfrlem8 37155 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑢) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
9895, 96, 973eqtr3d 2693 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
9910, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 90, 91, 92, 93, 94, 98lcfl7lem 37105 . . . 4 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 = 𝑢)
10099ex 449 . . 3 ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
101100ralrimivva 3000 . 2 (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
102 dff1o6 6571 . 2 (𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}) ↔ (𝐽 Fn (𝑉 ∖ { 0 }) ∧ ran 𝐽 = (𝐶 ∖ {𝑄}) ∧ ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢)))
1037, 89, 101, 102syl3anbrc 1265 1 (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  {csn 4210  cmpt 4762  ran crn 5144   Fn wfn 5921  1-1-ontowf1o 5925  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LModclmod 18911  LFnlclfn 34662  LKerclk 34690  LDualcld 34728  HLchlt 34955  LHypclh 35588  DVecHcdvh 36684  ocHcoch 36953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cntz 17796  df-lsm 18097  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151  df-lsatoms 34581  df-lshyp 34582  df-lfl 34663  df-lkr 34691  df-ldual 34729  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tgrp 36348  df-tendo 36360  df-edring 36362  df-dveca 36608  df-disoa 36635  df-dvech 36685  df-dib 36745  df-dic 36779  df-dih 36835  df-doch 36954  df-djh 37001
This theorem is referenced by:  lcf1o  37157
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