Theorem lcfrlem9 39564

Description: Lemma for lcf1o 39565. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 39565.) 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 6788	. . . . 5 ⊢ 𝑉 ∈ V
3	2	mptex 7099	. . . 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 6576	. . 3 ⊢ 𝐽 Fn (𝑉 ∖ { 0 })
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑉 ∖ { 0 }))
7		fvelrnb 6830	. . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24		simpr 485	. . . . . . . . 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 39563	. . . . . . . 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 4571	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
28	27	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ( ⊥ ‘{𝑦}) = ( ⊥ ‘{𝑧}))
29		oveq2 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
30	29	oveq2d 7291	. . . . . . . . . . . . . . . . 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 3337	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
33	32	riotabidv 7234	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
34	33	mpteq2dv 5176	. . . . . . . . . . . . 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 3575	. . . . . . . . . . . 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 871	. . . . . . . . . 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 39489	. . . . . . . . . . 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 39514	. . . . . . . . . 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 39488	. . . . . . . . . . . 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 39485	. . . . . . . . . . 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 39483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
44	42, 43	eqnetrrd 3012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
45	9, 11, 22	dvhlmod 39124	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	45	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
47	1, 17, 18, 19, 20, 46, 38	lkr0f2 37175	. . . . . . . . . . 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 2988	. . . . . . . . . 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 4720	. . . . . . . . 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 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 3213	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
56		eldifsn 4720	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄))
57		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐶)
58	45	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑈 ∈ LMod)
59	21	lcfl1lem 39505	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ 𝐶 ↔ (𝑔 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑔))) = (𝐿‘𝑔)))
60	59	simplbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐶 → 𝑔 ∈ 𝐹)
61	60	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ 𝐹)
62	1, 17, 18, 19, 20, 58, 61	lkr0f2 37175	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = 𝑄))
63	62	necon3bid 2988	. . . . . . . . . . . 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 455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → (𝐿‘𝑔) ≠ 𝑉)
66	65	neneqd 2948	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → ¬ (𝐿‘𝑔) = 𝑉)
67	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
68	60	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄) → 𝑔 ∈ 𝐹)
69	68	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐹)
70	9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 21, 67, 69	lcfl6 39514	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 861	. . . . . . . . . 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 2745	. . . . . . . . 9 ⊢ ((𝐽‘𝑧) = 𝑔 ↔ 𝑔 = (𝐽‘𝑧))
77	22	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
78		simpr 485	. . . . . . . . . . 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 39563	. . . . . . . . . 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 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 3225	. . . . . . 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 413	. . . . 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 723	. . . . 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 774	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
92		simplrr 775	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑢 ∈ (𝑉 ∖ { 0 }))
93		simpr 485	. . . . . 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 39563	. . . . . 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 39563	. . . . . 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 39513	. . . 4 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → 𝑡 = 𝑢)
98	97	ex 413	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) → ((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
99	98	ralrimivva 3123	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
100		dff1o6 7147	. 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 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884  {csn 4561   ↦ cmpt 5157  ran crn 5590   Fn wfn 6428  –1-1-onto→wf1o 6432  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  LModclmod 20123  LFnlclfn 37071  LKerclk 37099  LDualcld 37137  HLchlt 37364  LHypclh 37998  DVecHcdvh 39092  ocHcoch 39361

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-undef 8089  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-lmod 20125  df-lss 20194  df-lsp 20234  df-lvec 20365  df-lsatoms 36990  df-lshyp 36991  df-lfl 37072  df-lkr 37100  df-ldual 37138  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-tgrp 38757  df-tendo 38769  df-edring 38771  df-dveca 39017  df-disoa 39043  df-dvech 39093  df-dib 39153  df-dic 39187  df-dih 39243  df-doch 39362  df-djh 39409

This theorem is referenced by:  lcf1o  39565