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Theorem lcfrlem9 38238
Description: Lemma for lcf1o 38239. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 38239.) 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)
21fvexi 6559 . . . . 5 𝑉 ∈ V
32mptex 6859 . . . 4 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∈ V
4 lcf1o.j . . . 4 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
53, 4fnmpti 6366 . . 3 𝐽 Fn (𝑉 ∖ { 0 })
65a1i 11 . 2 (𝜑𝐽 Fn (𝑉 ∖ { 0 }))
7 fvelrnb 6601 . . . . 5 (𝐽 Fn (𝑉 ∖ { 0 }) → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
86, 7syl 17 . . . 4 (𝜑 → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
9 lcf1o.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
10 lcf1o.o . . . . . . . . 9 = ((ocH‘𝐾)‘𝑊)
11 lcf1o.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
12 lcf1o.a . . . . . . . . 9 + = (+g𝑈)
13 lcf1o.t . . . . . . . . 9 · = ( ·𝑠𝑈)
14 lcf1o.s . . . . . . . . 9 𝑆 = (Scalar‘𝑈)
15 lcf1o.r . . . . . . . . 9 𝑅 = (Base‘𝑆)
16 lcf1o.z . . . . . . . . 9 0 = (0g𝑈)
17 lcf1o.f . . . . . . . . 9 𝐹 = (LFnl‘𝑈)
18 lcf1o.l . . . . . . . . 9 𝐿 = (LKer‘𝑈)
19 lcf1o.d . . . . . . . . 9 𝐷 = (LDual‘𝑈)
20 lcf1o.q . . . . . . . . 9 𝑄 = (0g𝐷)
21 lcf1o.c . . . . . . . . 9 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
22 lcflo.k . . . . . . . . . 10 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2322adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
24 simpr 485 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
259, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 23, 24lcfrlem8 38237 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26 eqid 2797 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27 sneq 4488 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2827fveq2d 6549 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ( ‘{𝑦}) = ( ‘{𝑧}))
29 oveq2 7031 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
3029oveq2d 7039 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
3130eqeq2d 2807 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
3228, 31rexeqbidv 3364 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3332riotabidv 6986 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3433mpteq2dv 5063 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
3534rspceeqv 3579 . . . . . . . . . . . 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)))))
3624, 26, 35sylancl 586 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
3736olcd 871 . . . . . . . . . 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))))))
389, 10, 11, 1, 16, 12, 13, 17, 14, 15, 26, 23, 24dochflcl 38163 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐹)
399, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 23, 38lcfl6 38188 . . . . . . . . . 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)))))))
4037, 39mpbird 258 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶)
419, 10, 11, 1, 16, 12, 13, 18, 14, 15, 26, 23, 24dochsnkr2cl 38162 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))) ∖ { 0 }))
429, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem3 38159 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) = (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
439, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem1 38157 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
4442, 43eqnetrrd 3054 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
459, 11, 22dvhlmod 37798 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ LMod)
4645adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
471, 17, 18, 19, 20, 46, 38lkr0f2 35849 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = 𝑄))
4847necon3bid 3030 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
4944, 48mpbid 233 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50 eldifsn 4632 . . . . . . . . 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))
5140, 49, 50sylanbrc 583 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
5225, 51eqeltrd 2885 . . . . . . 7 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) ∈ (𝐶 ∖ {𝑄}))
53 eleq1 2872 . . . . . . 7 ((𝐽𝑧) = 𝑔 → ((𝐽𝑧) ∈ (𝐶 ∖ {𝑄}) ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
5452, 53syl5ibcom 246 . . . . . 6 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
5554rexlimdva 3249 . . . . 5 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
56 eldifsn 4632 . . . . . . . 8 (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔𝐶𝑔𝑄))
57 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐶)
5845adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑈 ∈ LMod)
5921lcfl1lem 38179 . . . . . . . . . . . . . . . 16 (𝑔𝐶 ↔ (𝑔𝐹 ∧ ( ‘( ‘(𝐿𝑔))) = (𝐿𝑔)))
6059simplbi 498 . . . . . . . . . . . . . . 15 (𝑔𝐶𝑔𝐹)
6160adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔𝐹)
621, 17, 18, 19, 20, 58, 61lkr0f2 35849 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → ((𝐿𝑔) = 𝑉𝑔 = 𝑄))
6362necon3bid 3030 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → ((𝐿𝑔) ≠ 𝑉𝑔𝑄))
6463biimprd 249 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → (𝑔𝑄 → (𝐿𝑔) ≠ 𝑉))
6564impr 455 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐿𝑔) ≠ 𝑉)
6665neneqd 2991 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ¬ (𝐿𝑔) = 𝑉)
6722adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6860adantr 481 . . . . . . . . . . . . . 14 ((𝑔𝐶𝑔𝑄) → 𝑔𝐹)
6968adantl 482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐹)
709, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69lcfl6 38188 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝑔𝐶 ↔ ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))))
7170biimpa 477 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
7271ord 859 . . . . . . . . . 10 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → (¬ (𝐿𝑔) = 𝑉 → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
73723impia 1110 . . . . . . . . 9 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶 ∧ ¬ (𝐿𝑔) = 𝑉) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7457, 66, 73mpd3an23 1455 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7556, 74sylan2b 593 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76 eqcom 2804 . . . . . . . . 9 ((𝐽𝑧) = 𝑔𝑔 = (𝐽𝑧))
7722ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
78 simpr 485 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
799, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 77, 78lcfrlem8 38237 . . . . . . . . . 10 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
8079eqeq2d 2807 . . . . . . . . 9 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑔 = (𝐽𝑧) ↔ 𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8176, 80syl5bb 284 . . . . . . . 8 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8281rexbidva 3261 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8375, 82mpbird 258 . . . . . 6 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔)
8483ex 413 . . . . 5 (𝜑 → (𝑔 ∈ (𝐶 ∖ {𝑄}) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
8555, 84impbid 213 . . . 4 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
868, 85bitrd 280 . . 3 (𝜑 → (𝑔 ∈ ran 𝐽𝑔 ∈ (𝐶 ∖ {𝑄})))
8786eqrdv 2795 . 2 (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
8822ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
89 eqid 2797 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90 eqid 2797 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
91 simplrl 773 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92 simplrr 774 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
93 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑡) = (𝐽𝑢))
949, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 91lcfrlem8 38237 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑡) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))))
959, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 92lcfrlem8 38237 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑢) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
9693, 94, 953eqtr3d 2841 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
979, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 88, 89, 90, 91, 92, 96lcfl7lem 38187 . . . 4 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 = 𝑢)
9897ex 413 . . 3 ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
9998ralrimivva 3160 . 2 (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
100 dff1o6 6904 . 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)))
1016, 87, 99, 100syl3anbrc 1336 1 (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842   = wceq 1525  wcel 2083  wne 2986  wral 3107  wrex 3108  {crab 3111  cdif 3862  {csn 4478  cmpt 5047  ran crn 5451   Fn wfn 6227  1-1-ontowf1o 6231  cfv 6232  crio 6983  (class class class)co 7023  Basecbs 16316  +gcplusg 16398  Scalarcsca 16401   ·𝑠 cvsca 16402  0gc0g 16546  LModclmod 19328  LFnlclfn 35745  LKerclk 35773  LDualcld 35811  HLchlt 36038  LHypclh 36672  DVecHcdvh 37766  ocHcoch 38035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-riotaBAD 35641
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-tpos 7750  df-undef 7797  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-n0 11752  df-z 11836  df-uz 12098  df-fz 12747  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-mulr 16412  df-sca 16414  df-vsca 16415  df-0g 16548  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-p1 17483  df-lat 17489  df-clat 17551  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-submnd 17779  df-grp 17868  df-minusg 17869  df-sbg 17870  df-subg 18034  df-cntz 18192  df-lsm 18495  df-cmn 18639  df-abl 18640  df-mgp 18934  df-ur 18946  df-ring 18993  df-oppr 19067  df-dvdsr 19085  df-unit 19086  df-invr 19116  df-dvr 19127  df-drng 19198  df-lmod 19330  df-lss 19398  df-lsp 19438  df-lvec 19569  df-lsatoms 35664  df-lshyp 35665  df-lfl 35746  df-lkr 35774  df-ldual 35812  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039  df-llines 36186  df-lplanes 36187  df-lvols 36188  df-lines 36189  df-psubsp 36191  df-pmap 36192  df-padd 36484  df-lhyp 36676  df-laut 36677  df-ldil 36792  df-ltrn 36793  df-trl 36847  df-tgrp 37431  df-tendo 37443  df-edring 37445  df-dveca 37691  df-disoa 37717  df-dvech 37767  df-dib 37827  df-dic 37861  df-dih 37917  df-doch 38036  df-djh 38083
This theorem is referenced by:  lcf1o  38239
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