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Theorem lcfrlem9 40063
Description: Lemma for lcf1o 40064. (This part has undesirable $d's on 𝐽 and πœ‘ that we remove in lcf1o 40064.) TODO: ugly proof; maybe have better subtheorems or abbreviate some β„©π‘˜ expansions with π½β€˜π‘§? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
Hypotheses
Ref Expression
lcf1o.h 𝐻 = (LHypβ€˜πΎ)
lcf1o.o βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
lcf1o.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
lcf1o.v 𝑉 = (Baseβ€˜π‘ˆ)
lcf1o.a + = (+gβ€˜π‘ˆ)
lcf1o.t Β· = ( ·𝑠 β€˜π‘ˆ)
lcf1o.s 𝑆 = (Scalarβ€˜π‘ˆ)
lcf1o.r 𝑅 = (Baseβ€˜π‘†)
lcf1o.z 0 = (0gβ€˜π‘ˆ)
lcf1o.f 𝐹 = (LFnlβ€˜π‘ˆ)
lcf1o.l 𝐿 = (LKerβ€˜π‘ˆ)
lcf1o.d 𝐷 = (LDualβ€˜π‘ˆ)
lcf1o.q 𝑄 = (0gβ€˜π·)
lcf1o.c 𝐢 = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}
lcf1o.j 𝐽 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))
lcflo.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
lcfrlem9 (πœ‘ β†’ 𝐽:(𝑉 βˆ– { 0 })–1-1-ontoβ†’(𝐢 βˆ– {𝑄}))
Distinct variable groups:   π‘₯,𝑀, βŠ₯   π‘₯, 0 ,𝑣   𝑣,𝑉,π‘₯   π‘₯, Β·   𝑣,π‘˜,𝑀,π‘₯, +   π‘₯,𝑅   𝑓,π‘˜,𝑣,𝑀,π‘₯, +   π‘˜,𝐽,𝑣,𝑀,π‘₯   𝐢,π‘˜,𝑣,𝑀,π‘₯   𝑓,𝐹   𝑓,𝐿,π‘˜,𝑣,𝑀,π‘₯   βŠ₯ ,𝑓,π‘˜,𝑣   𝑄,π‘˜,𝑣,𝑀,π‘₯   𝑅,𝑓,π‘˜,𝑣,𝑀   𝑆,π‘˜,𝑣,𝑀,π‘₯   Β· ,𝑓,π‘˜,𝑣,𝑀   π‘ˆ,π‘˜,𝑀,π‘₯   𝑓,𝑉,π‘˜,𝑀   0 ,π‘˜,𝑣,𝑀   πœ‘,π‘˜,𝑣,𝑀,π‘₯
Allowed substitution hints:   πœ‘(𝑓)   𝐢(𝑓)   𝐷(π‘₯,𝑀,𝑣,𝑓,π‘˜)   𝑄(𝑓)   𝑆(𝑓)   π‘ˆ(𝑣,𝑓)   𝐹(π‘₯,𝑀,𝑣,π‘˜)   𝐻(π‘₯,𝑀,𝑣,𝑓,π‘˜)   𝐽(𝑓)   𝐾(π‘₯,𝑀,𝑣,𝑓,π‘˜)   π‘Š(π‘₯,𝑀,𝑣,𝑓,π‘˜)   0 (𝑓)

Proof of Theorem lcfrlem9
Dummy variables 𝑦 𝑔 𝑑 𝑒 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcf1o.v . . . . . 6 𝑉 = (Baseβ€˜π‘ˆ)
21fvexi 6860 . . . . 5 𝑉 ∈ V
32mptex 7177 . . . 4 (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))) ∈ V
4 lcf1o.j . . . 4 𝐽 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))
53, 4fnmpti 6648 . . 3 𝐽 Fn (𝑉 βˆ– { 0 })
65a1i 11 . 2 (πœ‘ β†’ 𝐽 Fn (𝑉 βˆ– { 0 }))
7 fvelrnb 6907 . . . . 5 (𝐽 Fn (𝑉 βˆ– { 0 }) β†’ (𝑔 ∈ ran 𝐽 ↔ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔))
86, 7syl 17 . . . 4 (πœ‘ β†’ (𝑔 ∈ ran 𝐽 ↔ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔))
9 lcf1o.h . . . . . . . . 9 𝐻 = (LHypβ€˜πΎ)
10 lcf1o.o . . . . . . . . 9 βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
11 lcf1o.u . . . . . . . . 9 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
12 lcf1o.a . . . . . . . . 9 + = (+gβ€˜π‘ˆ)
13 lcf1o.t . . . . . . . . 9 Β· = ( ·𝑠 β€˜π‘ˆ)
14 lcf1o.s . . . . . . . . 9 𝑆 = (Scalarβ€˜π‘ˆ)
15 lcf1o.r . . . . . . . . 9 𝑅 = (Baseβ€˜π‘†)
16 lcf1o.z . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
17 lcf1o.f . . . . . . . . 9 𝐹 = (LFnlβ€˜π‘ˆ)
18 lcf1o.l . . . . . . . . 9 𝐿 = (LKerβ€˜π‘ˆ)
19 lcf1o.d . . . . . . . . 9 𝐷 = (LDualβ€˜π‘ˆ)
20 lcf1o.q . . . . . . . . 9 𝑄 = (0gβ€˜π·)
21 lcf1o.c . . . . . . . . 9 𝐢 = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}
22 lcflo.k . . . . . . . . . 10 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2322adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
24 simpr 486 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ 𝑧 ∈ (𝑉 βˆ– { 0 }))
259, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 23, 24lcfrlem8 40062 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (π½β€˜π‘§) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
26 eqid 2733 . . . . . . . . . . . 12 (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))
27 sneq 4600 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 β†’ {𝑦} = {𝑧})
2827fveq2d 6850 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 β†’ ( βŠ₯ β€˜{𝑦}) = ( βŠ₯ β€˜{𝑧}))
29 oveq2 7369 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 β†’ (π‘˜ Β· 𝑦) = (π‘˜ Β· 𝑧))
3029oveq2d 7377 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 β†’ (𝑀 + (π‘˜ Β· 𝑦)) = (𝑀 + (π‘˜ Β· 𝑧)))
3130eqeq2d 2744 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 β†’ (𝑣 = (𝑀 + (π‘˜ Β· 𝑦)) ↔ 𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))
3228, 31rexeqbidv 3319 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 β†’ (βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦)) ↔ βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))
3332riotabidv 7319 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 β†’ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦))) = (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))
3433mpteq2dv 5211 . . . . . . . . . . . . 13 (𝑦 = 𝑧 β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
3534rspceeqv 3599 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑉 βˆ– { 0 }) ∧ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) β†’ βˆƒπ‘¦ ∈ (𝑉 βˆ– { 0 })(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦)))))
3624, 26, 35sylancl 587 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ βˆƒπ‘¦ ∈ (𝑉 βˆ– { 0 })(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦)))))
3736olcd 873 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) = 𝑉 ∨ βˆƒπ‘¦ ∈ (𝑉 βˆ– { 0 })(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦))))))
389, 10, 11, 1, 16, 12, 13, 17, 14, 15, 26, 23, 24dochflcl 39988 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ 𝐹)
399, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 23, 38lcfl6 40013 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ 𝐢 ↔ ((πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) = 𝑉 ∨ βˆƒπ‘¦ ∈ (𝑉 βˆ– { 0 })(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑦})𝑣 = (𝑀 + (π‘˜ Β· 𝑦)))))))
4037, 39mpbird 257 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ 𝐢)
419, 10, 11, 1, 16, 12, 13, 18, 14, 15, 26, 23, 24dochsnkr2cl 39987 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ 𝑧 ∈ (( βŠ₯ β€˜(πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))) βˆ– { 0 }))
429, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem3 39984 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))) = (πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
439, 10, 11, 1, 16, 17, 18, 23, 38, 41dochsnkrlem1 39982 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))) β‰  𝑉)
4442, 43eqnetrrd 3009 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) β‰  𝑉)
459, 11, 22dvhlmod 39623 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LMod)
4645adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ π‘ˆ ∈ LMod)
471, 17, 18, 19, 20, 46, 38lkr0f2 37673 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) = 𝑉 ↔ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) = 𝑄))
4847necon3bid 2985 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((πΏβ€˜(𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))) β‰  𝑉 ↔ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) β‰  𝑄))
4944, 48mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) β‰  𝑄)
50 eldifsn 4751 . . . . . . . . 9 ((𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ (𝐢 βˆ– {𝑄}) ↔ ((𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ 𝐢 ∧ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) β‰  𝑄))
5140, 49, 50sylanbrc 584 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))) ∈ (𝐢 βˆ– {𝑄}))
5225, 51eqeltrd 2834 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (π½β€˜π‘§) ∈ (𝐢 βˆ– {𝑄}))
53 eleq1 2822 . . . . . . 7 ((π½β€˜π‘§) = 𝑔 β†’ ((π½β€˜π‘§) ∈ (𝐢 βˆ– {𝑄}) ↔ 𝑔 ∈ (𝐢 βˆ– {𝑄})))
5452, 53syl5ibcom 244 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((π½β€˜π‘§) = 𝑔 β†’ 𝑔 ∈ (𝐢 βˆ– {𝑄})))
5554rexlimdva 3149 . . . . 5 (πœ‘ β†’ (βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔 β†’ 𝑔 ∈ (𝐢 βˆ– {𝑄})))
56 eldifsn 4751 . . . . . . . 8 (𝑔 ∈ (𝐢 βˆ– {𝑄}) ↔ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄))
57 simprl 770 . . . . . . . . 9 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ 𝑔 ∈ 𝐢)
5845adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑔 ∈ 𝐢) β†’ π‘ˆ ∈ LMod)
5921lcfl1lem 40004 . . . . . . . . . . . . . . . 16 (𝑔 ∈ 𝐢 ↔ (𝑔 ∈ 𝐹 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘”))) = (πΏβ€˜π‘”)))
6059simplbi 499 . . . . . . . . . . . . . . 15 (𝑔 ∈ 𝐢 β†’ 𝑔 ∈ 𝐹)
6160adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑔 ∈ 𝐢) β†’ 𝑔 ∈ 𝐹)
621, 17, 18, 19, 20, 58, 61lkr0f2 37673 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑔 ∈ 𝐢) β†’ ((πΏβ€˜π‘”) = 𝑉 ↔ 𝑔 = 𝑄))
6362necon3bid 2985 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑔 ∈ 𝐢) β†’ ((πΏβ€˜π‘”) β‰  𝑉 ↔ 𝑔 β‰  𝑄))
6463biimprd 248 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 ∈ 𝐢) β†’ (𝑔 β‰  𝑄 β†’ (πΏβ€˜π‘”) β‰  𝑉))
6564impr 456 . . . . . . . . . 10 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ (πΏβ€˜π‘”) β‰  𝑉)
6665neneqd 2945 . . . . . . . . 9 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ Β¬ (πΏβ€˜π‘”) = 𝑉)
6722adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
6860adantr 482 . . . . . . . . . . . . . 14 ((𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄) β†’ 𝑔 ∈ 𝐹)
6968adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ 𝑔 ∈ 𝐹)
709, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69lcfl6 40013 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ (𝑔 ∈ 𝐢 ↔ ((πΏβ€˜π‘”) = 𝑉 ∨ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))))
7170biimpa 478 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) ∧ 𝑔 ∈ 𝐢) β†’ ((πΏβ€˜π‘”) = 𝑉 ∨ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
7271ord 863 . . . . . . . . . 10 (((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) ∧ 𝑔 ∈ 𝐢) β†’ (Β¬ (πΏβ€˜π‘”) = 𝑉 β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
73723impia 1118 . . . . . . . . 9 (((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) ∧ 𝑔 ∈ 𝐢 ∧ Β¬ (πΏβ€˜π‘”) = 𝑉) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
7457, 66, 73mpd3an23 1464 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ 𝐢 ∧ 𝑔 β‰  𝑄)) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
7556, 74sylan2b 595 . . . . . . 7 ((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
76 eqcom 2740 . . . . . . . . 9 ((π½β€˜π‘§) = 𝑔 ↔ 𝑔 = (π½β€˜π‘§))
7722ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
78 simpr 486 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ 𝑧 ∈ (𝑉 βˆ– { 0 }))
799, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 77, 78lcfrlem8 40062 . . . . . . . . . 10 (((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (π½β€˜π‘§) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧)))))
8079eqeq2d 2744 . . . . . . . . 9 (((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ (𝑔 = (π½β€˜π‘§) ↔ 𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
8176, 80bitrid 283 . . . . . . . 8 (((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) ∧ 𝑧 ∈ (𝑉 βˆ– { 0 })) β†’ ((π½β€˜π‘§) = 𝑔 ↔ 𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
8281rexbidva 3170 . . . . . . 7 ((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) β†’ (βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔 ↔ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑧})𝑣 = (𝑀 + (π‘˜ Β· 𝑧))))))
8375, 82mpbird 257 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ (𝐢 βˆ– {𝑄})) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔)
8483ex 414 . . . . 5 (πœ‘ β†’ (𝑔 ∈ (𝐢 βˆ– {𝑄}) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔))
8555, 84impbid 211 . . . 4 (πœ‘ β†’ (βˆƒπ‘§ ∈ (𝑉 βˆ– { 0 })(π½β€˜π‘§) = 𝑔 ↔ 𝑔 ∈ (𝐢 βˆ– {𝑄})))
868, 85bitrd 279 . . 3 (πœ‘ β†’ (𝑔 ∈ ran 𝐽 ↔ 𝑔 ∈ (𝐢 βˆ– {𝑄})))
8786eqrdv 2731 . 2 (πœ‘ β†’ ran 𝐽 = (𝐢 βˆ– {𝑄}))
8822ad2antrr 725 . . . . 5 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
89 eqid 2733 . . . . 5 (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑑})𝑣 = (𝑀 + (π‘˜ Β· 𝑑)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑑})𝑣 = (𝑀 + (π‘˜ Β· 𝑑))))
90 eqid 2733 . . . . 5 (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑒})𝑣 = (𝑀 + (π‘˜ Β· 𝑒)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑒})𝑣 = (𝑀 + (π‘˜ Β· 𝑒))))
91 simplrl 776 . . . . 5 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ 𝑑 ∈ (𝑉 βˆ– { 0 }))
92 simplrr 777 . . . . 5 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ 𝑒 ∈ (𝑉 βˆ– { 0 }))
93 simpr 486 . . . . . 6 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ (π½β€˜π‘‘) = (π½β€˜π‘’))
949, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 91lcfrlem8 40062 . . . . . 6 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ (π½β€˜π‘‘) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑑})𝑣 = (𝑀 + (π‘˜ Β· 𝑑)))))
959, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 92lcfrlem8 40062 . . . . . 6 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ (π½β€˜π‘’) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑒})𝑣 = (𝑀 + (π‘˜ Β· 𝑒)))))
9693, 94, 953eqtr3d 2781 . . . . 5 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑑})𝑣 = (𝑀 + (π‘˜ Β· 𝑑)))) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑒})𝑣 = (𝑀 + (π‘˜ Β· 𝑒)))))
979, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 88, 89, 90, 91, 92, 96lcfl7lem 40012 . . . 4 (((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) ∧ (π½β€˜π‘‘) = (π½β€˜π‘’)) β†’ 𝑑 = 𝑒)
9897ex 414 . . 3 ((πœ‘ ∧ (𝑑 ∈ (𝑉 βˆ– { 0 }) ∧ 𝑒 ∈ (𝑉 βˆ– { 0 }))) β†’ ((π½β€˜π‘‘) = (π½β€˜π‘’) β†’ 𝑑 = 𝑒))
9998ralrimivva 3194 . 2 (πœ‘ β†’ βˆ€π‘‘ ∈ (𝑉 βˆ– { 0 })βˆ€π‘’ ∈ (𝑉 βˆ– { 0 })((π½β€˜π‘‘) = (π½β€˜π‘’) β†’ 𝑑 = 𝑒))
100 dff1o6 7225 . 2 (𝐽:(𝑉 βˆ– { 0 })–1-1-ontoβ†’(𝐢 βˆ– {𝑄}) ↔ (𝐽 Fn (𝑉 βˆ– { 0 }) ∧ ran 𝐽 = (𝐢 βˆ– {𝑄}) ∧ βˆ€π‘‘ ∈ (𝑉 βˆ– { 0 })βˆ€π‘’ ∈ (𝑉 βˆ– { 0 })((π½β€˜π‘‘) = (π½β€˜π‘’) β†’ 𝑑 = 𝑒)))
1016, 87, 99, 100syl3anbrc 1344 1 (πœ‘ β†’ 𝐽:(𝑉 βˆ– { 0 })–1-1-ontoβ†’(𝐢 βˆ– {𝑄}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   βˆ– cdif 3911  {csn 4590   ↦ cmpt 5192  ran crn 5638   Fn wfn 6495  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  Scalarcsca 17144   ·𝑠 cvsca 17145  0gc0g 17329  LModclmod 20365  LFnlclfn 37569  LKerclk 37597  LDualcld 37635  HLchlt 37862  LHypclh 38497  DVecHcdvh 39591  ocHcoch 39860
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-riotaBAD 37465
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-undef 8208  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-0g 17331  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-cntz 19105  df-lsm 19426  df-cmn 19572  df-abl 19573  df-mgp 19905  df-ur 19922  df-ring 19974  df-oppr 20057  df-dvdsr 20078  df-unit 20079  df-invr 20109  df-dvr 20120  df-drng 20221  df-lmod 20367  df-lss 20437  df-lsp 20477  df-lvec 20608  df-lsatoms 37488  df-lshyp 37489  df-lfl 37570  df-lkr 37598  df-ldual 37636  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-llines 38011  df-lplanes 38012  df-lvols 38013  df-lines 38014  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-tgrp 39256  df-tendo 39268  df-edring 39270  df-dveca 39516  df-disoa 39542  df-dvech 39592  df-dib 39652  df-dic 39686  df-dih 39742  df-doch 39861  df-djh 39908
This theorem is referenced by:  lcf1o  40064
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