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Theorem lcfrlem9 40327
Description: Lemma for lcf1o 40328. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 40328.) 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 6895 . . . . 5 𝑉 ∈ V
32mptex 7212 . . . 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 6683 . . 3 𝐽 Fn (𝑉 ∖ { 0 })
65a1i 11 . 2 (𝜑𝐽 Fn (𝑉 ∖ { 0 }))
7 fvelrnb 6942 . . . . 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 482 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
24 simpr 486 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
259, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 23, 24lcfrlem8 40326 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26 eqid 2733 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27 sneq 4634 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2827fveq2d 6885 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ( ‘{𝑦}) = ( ‘{𝑧}))
29 oveq2 7404 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
3029oveq2d 7412 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
3130eqeq2d 2744 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
3228, 31rexeqbidv 3344 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3332riotabidv 7354 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3433mpteq2dv 5246 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
3534rspceeqv 3631 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
3736olcd 873 . . . . . . . . . 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 40252 . . . . . . . . . . 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 40277 . . . . . . . . . 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 257 . . . . . . . . 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 40251 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))) ∖ { 0 }))
429, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem3 40248 . . . . . . . . . . 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 40246 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
4442, 43eqnetrrd 3010 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
459, 11, 22dvhlmod 39887 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ LMod)
4645adantr 482 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
471, 17, 18, 19, 20, 46, 38lkr0f2 37937 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = 𝑄))
4847necon3bid 2986 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
4944, 48mpbid 231 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50 eldifsn 4786 . . . . . . . . 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 584 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
5225, 51eqeltrd 2834 . . . . . . 7 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) ∈ (𝐶 ∖ {𝑄}))
53 eleq1 2822 . . . . . . 7 ((𝐽𝑧) = 𝑔 → ((𝐽𝑧) ∈ (𝐶 ∖ {𝑄}) ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
5452, 53syl5ibcom 244 . . . . . 6 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
5554rexlimdva 3156 . . . . 5 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
56 eldifsn 4786 . . . . . . . 8 (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔𝐶𝑔𝑄))
57 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐶)
5845adantr 482 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑈 ∈ LMod)
5921lcfl1lem 40268 . . . . . . . . . . . . . . . 16 (𝑔𝐶 ↔ (𝑔𝐹 ∧ ( ‘( ‘(𝐿𝑔))) = (𝐿𝑔)))
6059simplbi 499 . . . . . . . . . . . . . . 15 (𝑔𝐶𝑔𝐹)
6160adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔𝐹)
621, 17, 18, 19, 20, 58, 61lkr0f2 37937 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → ((𝐿𝑔) = 𝑉𝑔 = 𝑄))
6362necon3bid 2986 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → ((𝐿𝑔) ≠ 𝑉𝑔𝑄))
6463biimprd 247 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → (𝑔𝑄 → (𝐿𝑔) ≠ 𝑉))
6564impr 456 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐿𝑔) ≠ 𝑉)
6665neneqd 2946 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ¬ (𝐿𝑔) = 𝑉)
6722adantr 482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6860adantr 482 . . . . . . . . . . . . . 14 ((𝑔𝐶𝑔𝑄) → 𝑔𝐹)
6968adantl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐹)
709, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69lcfl6 40277 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝑔𝐶 ↔ ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))))
7170biimpa 478 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
7271ord 863 . . . . . . . . . 10 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → (¬ (𝐿𝑔) = 𝑉 → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
73723impia 1118 . . . . . . . . 9 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶 ∧ ¬ (𝐿𝑔) = 𝑉) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7457, 66, 73mpd3an23 1464 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7556, 74sylan2b 595 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76 eqcom 2740 . . . . . . . . 9 ((𝐽𝑧) = 𝑔𝑔 = (𝐽𝑧))
7722ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
78 simpr 486 . . . . . . . . . . 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 40326 . . . . . . . . . 10 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
8079eqeq2d 2744 . . . . . . . . 9 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑔 = (𝐽𝑧) ↔ 𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8176, 80bitrid 283 . . . . . . . 8 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8281rexbidva 3177 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8375, 82mpbird 257 . . . . . 6 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔)
8483ex 414 . . . . 5 (𝜑 → (𝑔 ∈ (𝐶 ∖ {𝑄}) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
8555, 84impbid 211 . . . 4 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
868, 85bitrd 279 . . 3 (𝜑 → (𝑔 ∈ ran 𝐽𝑔 ∈ (𝐶 ∖ {𝑄})))
8786eqrdv 2731 . 2 (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
8822ad2antrr 725 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
89 eqid 2733 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90 eqid 2733 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
91 simplrl 776 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92 simplrr 777 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
93 simpr 486 . . . . . 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 40326 . . . . . 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 40326 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑢) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
9693, 94, 953eqtr3d 2781 . . . . 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 40276 . . . 4 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 = 𝑢)
9897ex 414 . . 3 ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
9998ralrimivva 3201 . 2 (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
100 dff1o6 7260 . 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 1344 1 (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  cdif 3943  {csn 4624  cmpt 5227  ran crn 5673   Fn wfn 6530  1-1-ontowf1o 6534  cfv 6535  crio 7351  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  Scalarcsca 17187   ·𝑠 cvsca 17188  0gc0g 17372  LModclmod 20448  LFnlclfn 37833  LKerclk 37861  LDualcld 37899  HLchlt 38126  LHypclh 38761  DVecHcdvh 39855  ocHcoch 40124
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174  ax-riotaBAD 37729
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-1st 7962  df-2nd 7963  df-tpos 8198  df-undef 8245  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-3 12263  df-4 12264  df-5 12265  df-6 12266  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472  df-struct 17067  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-mulr 17198  df-sca 17200  df-vsca 17201  df-0g 17374  df-proset 18235  df-poset 18253  df-plt 18270  df-lub 18286  df-glb 18287  df-join 18288  df-meet 18289  df-p0 18365  df-p1 18366  df-lat 18372  df-clat 18439  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-submnd 18659  df-grp 18809  df-minusg 18810  df-sbg 18811  df-subg 18988  df-cntz 19166  df-lsm 19488  df-cmn 19634  df-abl 19635  df-mgp 19971  df-ur 19988  df-ring 20040  df-oppr 20128  df-dvdsr 20149  df-unit 20150  df-invr 20180  df-dvr 20193  df-drng 20295  df-lmod 20450  df-lss 20520  df-lsp 20560  df-lvec 20691  df-lsatoms 37752  df-lshyp 37753  df-lfl 37834  df-lkr 37862  df-ldual 37900  df-oposet 37952  df-ol 37954  df-oml 37955  df-covers 38042  df-ats 38043  df-atl 38074  df-cvlat 38098  df-hlat 38127  df-llines 38275  df-lplanes 38276  df-lvols 38277  df-lines 38278  df-psubsp 38280  df-pmap 38281  df-padd 38573  df-lhyp 38765  df-laut 38766  df-ldil 38881  df-ltrn 38882  df-trl 38936  df-tgrp 39520  df-tendo 39532  df-edring 39534  df-dveca 39780  df-disoa 39806  df-dvech 39856  df-dib 39916  df-dic 39950  df-dih 40006  df-doch 40125  df-djh 40172
This theorem is referenced by:  lcf1o  40328
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