Theorem lcfrlem9 41532

Description: Lemma for lcf1o 41533. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 41533.) TODO: ugly proof; maybe have better subtheorems or abbreviate some ℩𝑘 expansions with 𝐽‘𝑧? 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	⊢ + = (+g‘𝑈)
lcf1o.t	⊢ · = ( ·𝑠 ‘𝑈)
lcf1o.s	⊢ 𝑆 = (Scalar‘𝑈)
lcf1o.r	⊢ 𝑅 = (Base‘𝑆)
lcf1o.z	⊢ 0 = (0g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0g‘𝐷)
lcf1o.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcf1o.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
lcfrlem9	⊢ (𝜑 → 𝐽:(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.

Step	Hyp	Ref	Expression
1		lcf1o.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
2	1	fvexi 6920	. . . . 5 ⊢ 𝑉 ∈ V
3	2	mptex 7242	. . . 4 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) ∈ V
4		lcf1o.j	. . . 4 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
5	3, 4	fnmpti 6711	. . 3 ⊢ 𝐽 Fn (𝑉 ∖ { 0 })
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑉 ∖ { 0 }))
7		fvelrnb 6968	. . . . 5 ⊢ (𝐽 Fn (𝑉 ∖ { 0 }) → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝑔 ∈ ran 𝐽 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 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 ⊢ + = (+g‘𝑈)
13		lcf1o.t	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑈)
14		lcf1o.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑈)
15		lcf1o.r	. . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑆)
16		lcf1o.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
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 = (0g‘𝐷)
21		lcf1o.c	. . . . . . . . 9 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22		lcflo.k	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
25	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 23, 24	lcfrlem8 41531	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27		sneq 4640	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
28	27	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ( ⊥ ‘{𝑦}) = ( ⊥ ‘{𝑧}))
29		oveq2 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
30	29	oveq2d 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
31	30	eqeq2d 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
32	28, 31	rexeqbidv 3344	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
33	32	riotabidv 7389	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
34	33	mpteq2dv 5249	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
35	34	rspceeqv 3644	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
36	24, 26, 35	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))
37	36	olcd 874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ∨ ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))))))
38	9, 10, 11, 1, 16, 12, 13, 17, 14, 15, 26, 23, 24	dochflcl 41457	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐹)
39	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 23, 38	lcfl6 41482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶 ↔ ((𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ∨ ∃𝑦 ∈ (𝑉 ∖ { 0 })(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)))))))
40	37, 39	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶)
41	9, 10, 11, 1, 16, 12, 13, 18, 14, 15, 26, 23, 24	dochsnkr2cl 41456	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))) ∖ { 0 }))
42	9, 10, 11, 1, 16, 17, 18, 23, 38, 41	dochsnkrlem3 41453	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) = (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
43	9, 10, 11, 1, 16, 17, 18, 23, 38, 41	dochsnkrlem1 41451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
44	42, 43	eqnetrrd 3006	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
45	9, 11, 22	dvhlmod 41092	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
47	1, 17, 18, 19, 20, 46, 38	lkr0f2 39142	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) = 𝑉 ↔ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = 𝑄))
48	47	necon3bid 2982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉 ↔ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
49	44, 48	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50		eldifsn 4790	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}) ↔ ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ 𝐶 ∧ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄))
51	40, 49, 50	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
52	25, 51	eqeltrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) ∈ (𝐶 ∖ {𝑄}))
53		eleq1 2826	. . . . . . 7 ⊢ ((𝐽‘𝑧) = 𝑔 → ((𝐽‘𝑧) ∈ (𝐶 ∖ {𝑄}) ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
54	52, 53	syl5ibcom 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
55	54	rexlimdva 3152	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
56		eldifsn 4790	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄))
57		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐶)
58	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑈 ∈ LMod)
59	21	lcfl1lem 41473	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ 𝐶 ↔ (𝑔 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑔))) = (𝐿‘𝑔)))
60	59	simplbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐶 → 𝑔 ∈ 𝐹)
61	60	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ 𝐹)
62	1, 17, 18, 19, 20, 58, 61	lkr0f2 39142	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = 𝑄))
63	62	necon3bid 2982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) ≠ 𝑉 ↔ 𝑔 ≠ 𝑄))
64	63	biimprd 248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → (𝑔 ≠ 𝑄 → (𝐿‘𝑔) ≠ 𝑉))
65	64	impr 454	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → (𝐿‘𝑔) ≠ 𝑉)
66	65	neneqd 2942	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → ¬ (𝐿‘𝑔) = 𝑉)
67	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
68	60	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄) → 𝑔 ∈ 𝐹)
69	68	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐹)
70	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69	lcfl6 41482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → (𝑔 ∈ 𝐶 ↔ ((𝐿‘𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))))
71	70	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) = 𝑉 ∨ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
72	71	ord 864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) ∧ 𝑔 ∈ 𝐶) → (¬ (𝐿‘𝑔) = 𝑉 → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
73	72	3impia 1116	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) ∧ 𝑔 ∈ 𝐶 ∧ ¬ (𝐿‘𝑔) = 𝑉) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
74	57, 66, 73	mpd3an23 1462	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
75	56, 74	sylan2b 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76		eqcom 2741	. . . . . . . . 9 ⊢ ((𝐽‘𝑧) = 𝑔 ↔ 𝑔 = (𝐽‘𝑧))
77	22	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
78		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑧 ∈ (𝑉 ∖ { 0 }))
79	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 77, 78	lcfrlem8 41531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
80	79	eqeq2d 2745	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑔 = (𝐽‘𝑧) ↔ 𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
81	76, 80	bitrid 283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽‘𝑧) = 𝑔 ↔ 𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
82	81	rexbidva 3174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 ↔ ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))
83	75, 82	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔)
84	83	ex 412	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (𝐶 ∖ {𝑄}) → ∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔))
85	55, 84	impbid 212	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
86	8, 85	bitrd 279	. . 3 ⊢ (𝜑 → (𝑔 ∈ ran 𝐽 ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
87	86	eqrdv 2732	. 2 ⊢ (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
88	22	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
89		eqid 2734	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90		eqid 2734	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
91		simplrl 777	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92		simplrr 778	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
93		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐽‘𝑡) = (𝐽‘𝑢))
94	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 91	lcfrlem8 41531	. . . . . 6 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐽‘𝑡) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))))
95	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 4, 88, 92	lcfrlem8 41531	. . . . . 6 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐽‘𝑢) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
96	93, 94, 95	3eqtr3d 2782	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))))
97	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 88, 89, 90, 91, 92, 96	lcfl7lem 41481	. . . 4 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑡 = 𝑢)
98	97	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
99	98	ralrimivva 3199	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
100		dff1o6 7294	. 2 ⊢ (𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}) ↔ (𝐽 Fn (𝑉 ∖ { 0 }) ∧ ran 𝐽 = (𝐶 ∖ {𝑄}) ∧ ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢)))
101	6, 87, 99, 100	syl3anbrc 1342	1 ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3432   ∖ cdif 3959  {csn 4630   ↦ cmpt 5230  ran crn 5689   Fn wfn 6557  –1-1-onto→wf1o 6561  ‘cfv 6562  ℩crio 7386  (class class class)co 7430  Basecbs 17244  +_gcplusg 17297  Scalarcsca 17300   ·_𝑠 cvsca 17301  0_gc0g 17485  LModclmod 20874  LFnlclfn 39038  LKerclk 39066  LDualcld 39104  HLchlt 39331  LHypclh 39966  DVecHcdvh 41060  ocHcoch 41329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-riotaBAD 38934

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-undef 8296  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17487  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-p1 18483  df-lat 18489  df-clat 18556  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-subg 19153  df-cntz 19347  df-lsm 19668  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-drng 20747  df-lmod 20876  df-lss 20947  df-lsp 20987  df-lvec 21119  df-lsatoms 38957  df-lshyp 38958  df-lfl 39039  df-lkr 39067  df-ldual 39105  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-llines 39480  df-lplanes 39481  df-lvols 39482  df-lines 39483  df-psubsp 39485  df-pmap 39486  df-padd 39778  df-lhyp 39970  df-laut 39971  df-ldil 40086  df-ltrn 40087  df-trl 40141  df-tgrp 40725  df-tendo 40737  df-edring 40739  df-dveca 40985  df-disoa 41011  df-dvech 41061  df-dib 41121  df-dic 41155  df-dih 41211  df-doch 41330  df-djh 41377

This theorem is referenced by:  lcf1o  41533