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Theorem lcfrlem9 37506
Description: Lemma for lcf1o 37507. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 37507.) 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 6389 . . . . 5 𝑉 ∈ V
32mptex 6679 . . . 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 6200 . . 3 𝐽 Fn (𝑉 ∖ { 0 })
65a1i 11 . 2 (𝜑𝐽 Fn (𝑉 ∖ { 0 }))
7 fvelrnb 6432 . . . . 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 472 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
24 simpr 477 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
259, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 23, 24lcfrlem8 37505 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26 eqid 2765 . . . . . . . . . . . 12 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27 sneq 4344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2827fveq2d 6379 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ( ‘{𝑦}) = ( ‘{𝑧}))
29 oveq2 6850 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
3029oveq2d 6858 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
3130eqeq2d 2775 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
3228, 31rexeqbidv 3301 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (∃𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3332riotabidv 6805 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
3433mpteq2dv 4904 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
3534rspceeqv 3479 . . . . . . . . . . . 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 580 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
3736olcd 900 . . . . . . . . . 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 37431 . . . . . . . . . . 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 37456 . . . . . . . . . 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 248 . . . . . . . . 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 37430 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))) ∖ { 0 }))
429, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem3 37427 . . . . . . . . . . 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 37425 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ( ‘( ‘(𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
4442, 43eqnetrrd 3005 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
459, 11, 22dvhlmod 37066 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ LMod)
4645adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
471, 17, 18, 19, 20, 46, 38lkr0f2 35117 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = 𝑄))
4847necon3bid 2981 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉 ↔ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
4944, 48mpbid 223 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50 eldifsn 4472 . . . . . . . . 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 578 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
5225, 51eqeltrd 2844 . . . . . . 7 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) ∈ (𝐶 ∖ {𝑄}))
53 eleq1 2832 . . . . . . 7 ((𝐽𝑧) = 𝑔 → ((𝐽𝑧) ∈ (𝐶 ∖ {𝑄}) ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
5452, 53syl5ibcom 236 . . . . . 6 ((𝜑𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
5554rexlimdva 3178 . . . . 5 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
56 eldifsn 4472 . . . . . . . 8 (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔𝐶𝑔𝑄))
57 simprl 787 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐶)
5845adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑈 ∈ LMod)
5921lcfl1lem 37447 . . . . . . . . . . . . . . . 16 (𝑔𝐶 ↔ (𝑔𝐹 ∧ ( ‘( ‘(𝐿𝑔))) = (𝐿𝑔)))
6059simplbi 491 . . . . . . . . . . . . . . 15 (𝑔𝐶𝑔𝐹)
6160adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔𝐹)
621, 17, 18, 19, 20, 58, 61lkr0f2 35117 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → ((𝐿𝑔) = 𝑉𝑔 = 𝑄))
6362necon3bid 2981 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → ((𝐿𝑔) ≠ 𝑉𝑔𝑄))
6463biimprd 239 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → (𝑔𝑄 → (𝐿𝑔) ≠ 𝑉))
6564impr 446 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐿𝑔) ≠ 𝑉)
6665neneqd 2942 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ¬ (𝐿𝑔) = 𝑉)
6722adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6860adantr 472 . . . . . . . . . . . . . 14 ((𝑔𝐶𝑔𝑄) → 𝑔𝐹)
6968adantl 473 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → 𝑔𝐹)
709, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69lcfl6 37456 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → (𝑔𝐶 ↔ ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))))
7170biimpa 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → ((𝐿𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
7271ord 890 . . . . . . . . . 10 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶) → (¬ (𝐿𝑔) = 𝑉 → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
73723impia 1145 . . . . . . . . 9 (((𝜑 ∧ (𝑔𝐶𝑔𝑄)) ∧ 𝑔𝐶 ∧ ¬ (𝐿𝑔) = 𝑉) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7457, 66, 73mpd3an23 1587 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐶𝑔𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
7556, 74sylan2b 587 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76 eqcom 2772 . . . . . . . . 9 ((𝐽𝑧) = 𝑔𝑔 = (𝐽𝑧))
7722ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
78 simpr 477 . . . . . . . . . . 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 37505 . . . . . . . . . 10 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽𝑧) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
8079eqeq2d 2775 . . . . . . . . 9 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑔 = (𝐽𝑧) ↔ 𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8176, 80syl5bb 274 . . . . . . . 8 (((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽𝑧) = 𝑔𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8281rexbidva 3196 . . . . . . 7 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
8375, 82mpbird 248 . . . . . 6 ((𝜑𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔)
8483ex 401 . . . . 5 (𝜑 → (𝑔 ∈ (𝐶 ∖ {𝑄}) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔))
8555, 84impbid 203 . . . 4 (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽𝑧) = 𝑔𝑔 ∈ (𝐶 ∖ {𝑄})))
868, 85bitrd 270 . . 3 (𝜑 → (𝑔 ∈ ran 𝐽𝑔 ∈ (𝐶 ∖ {𝑄})))
8786eqrdv 2763 . 2 (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
8822ad2antrr 717 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
89 eqid 2765 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90 eqid 2765 . . . . 5 (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
91 simplrl 795 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92 simplrr 796 . . . . 5 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
93 simpr 477 . . . . . 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 37505 . . . . . 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 37505 . . . . . 6 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → (𝐽𝑢) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
9693, 94, 953eqtr3d 2807 . . . . 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 37455 . . . 4 (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽𝑡) = (𝐽𝑢)) → 𝑡 = 𝑢)
9897ex 401 . . 3 ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
9998ralrimivva 3118 . 2 (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽𝑡) = (𝐽𝑢) → 𝑡 = 𝑢))
100 dff1o6 6723 . 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 1443 1 (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cdif 3729  {csn 4334  cmpt 4888  ran crn 5278   Fn wfn 6063  1-1-ontowf1o 6067  cfv 6068  crio 6802  (class class class)co 6842  Basecbs 16130  +gcplusg 16214  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366  LModclmod 19132  LFnlclfn 35013  LKerclk 35041  LDualcld 35079  HLchlt 35306  LHypclh 35940  DVecHcdvh 37034  ocHcoch 37303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-riotaBAD 34909
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-tpos 7555  df-undef 7602  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-0g 16368  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-p1 17306  df-lat 17312  df-clat 17374  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-subg 17855  df-cntz 18013  df-lsm 18315  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-dvr 18950  df-drng 19018  df-lmod 19134  df-lss 19202  df-lsp 19244  df-lvec 19375  df-lsatoms 34932  df-lshyp 34933  df-lfl 35014  df-lkr 35042  df-ldual 35080  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307  df-llines 35454  df-lplanes 35455  df-lvols 35456  df-lines 35457  df-psubsp 35459  df-pmap 35460  df-padd 35752  df-lhyp 35944  df-laut 35945  df-ldil 36060  df-ltrn 36061  df-trl 36115  df-tgrp 36699  df-tendo 36711  df-edring 36713  df-dveca 36959  df-disoa 36985  df-dvech 37035  df-dib 37095  df-dic 37129  df-dih 37185  df-doch 37304  df-djh 37351
This theorem is referenced by:  lcf1o  37507
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