Theorem lcfrlem9 39491

Description: Lemma for lcf1o 39492. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 39492.) 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 6770	. . . . 5 ⊢ 𝑉 ∈ V
3	2	mptex 7081	. . . 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 6560	. . 3 ⊢ 𝐽 Fn (𝑉 ∖ { 0 })
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑉 ∖ { 0 }))
7		fvelrnb 6812	. . . . 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 39490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27		sneq 4568	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
28	27	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ( ⊥ ‘{𝑦}) = ( ⊥ ‘{𝑧}))
29		oveq2 7263	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
30	29	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
31	30	eqeq2d 2749	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
32	28, 31	rexeqbidv 3328	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
33	32	riotabidv 7214	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
34	33	mpteq2dv 5172	. . . . . . . . . . . . 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 3567	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 870	. . . . . . . . . 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 39416	. . . . . . . . . . 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 39441	. . . . . . . . . 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 256	. . . . . . . . 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 39415	. . . . . . . . . . . 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 39412	. . . . . . . . . . 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 39410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
44	42, 43	eqnetrrd 3011	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
45	9, 11, 22	dvhlmod 39051	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
47	1, 17, 18, 19, 20, 46, 38	lkr0f2 37102	. . . . . . . . . . 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 2987	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50		eldifsn 4717	. . . . . . . . 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 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
52	25, 51	eqeltrd 2839	. . . . . . 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 244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ((𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
55	54	rexlimdva 3212	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
56		eldifsn 4717	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄))
57		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐶)
58	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑈 ∈ LMod)
59	21	lcfl1lem 39432	. . . . . . . . . . . . . . . 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 37102	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = 𝑄))
63	62	necon3bid 2987	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) ≠ 𝑉 ↔ 𝑔 ≠ 𝑄))
64	63	biimprd 247	. . . . . . . . . . 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 2947	. . . . . . . . 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 39441	. . . . . . . . . . . 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 860	. . . . . . . . . 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 1115	. . . . . . . . 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 1461	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
75	56, 74	sylan2b 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76		eqcom 2745	. . . . . . . . 9 ⊢ ((𝐽‘𝑧) = 𝑔 ↔ 𝑔 = (𝐽‘𝑧))
77	22	ad2antrr 722	. . . . . . . . . . 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 39490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
80	79	eqeq2d 2749	. . . . . . . . 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	syl5bb 282	. . . . . . . 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 3224	. . . . . . 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 256	. . . . . 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 211	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
86	8, 85	bitrd 278	. . 3 ⊢ (𝜑 → (𝑔 ∈ ran 𝐽 ↔ 𝑔 ∈ (𝐶 ∖ {𝑄})))
87	86	eqrdv 2736	. 2 ⊢ (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
88	22	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
89		eqid 2738	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90		eqid 2738	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑢})𝑣 = (𝑤 + (𝑘 · 𝑢))))
91		simplrl 773	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92		simplrr 774	. . . . 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 39490	. . . . . 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 39490	. . . . . 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 2786	. . . . 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 39440	. . . 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 3114	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
100		dff1o6 7128	. 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 1341	1 ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880  {csn 4558   ↦ cmpt 5153  ran crn 5581   Fn wfn 6413  –1-1-onto→wf1o 6417  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  LModclmod 20038  LFnlclfn 36998  LKerclk 37026  LDualcld 37064  HLchlt 37291  LHypclh 37925  DVecHcdvh 39019  ocHcoch 39288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-undef 8060  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-0g 17069  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lvec 20280  df-lsatoms 36917  df-lshyp 36918  df-lfl 36999  df-lkr 37027  df-ldual 37065  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tgrp 38684  df-tendo 38696  df-edring 38698  df-dveca 38944  df-disoa 38970  df-dvech 39020  df-dib 39080  df-dic 39114  df-dih 39170  df-doch 39289  df-djh 39336

This theorem is referenced by:  lcf1o  39492