Theorem lcfrlem9 42007

Description: Lemma for lcf1o 42008. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 42008.) 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 6846	. . . . 5 ⊢ 𝑉 ∈ V
3	2	mptex 7169	. . . 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 6633	. . 3 ⊢ 𝐽 Fn (𝑉 ∖ { 0 })
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑉 ∖ { 0 }))
7		fvelrnb 6892	. . . . 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 42006	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27		sneq 4578	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
28	27	fveq2d 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ( ⊥ ‘{𝑦}) = ( ⊥ ‘{𝑧}))
29		oveq2 7366	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
30	29	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
31	30	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
32	28, 31	rexeqbidv 3313	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
33	32	riotabidv 7317	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
34	33	mpteq2dv 5180	. . . . . . . . . . . . 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 3588	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 875	. . . . . . . . . 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 41932	. . . . . . . . . . 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 41957	. . . . . . . . . 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 41931	. . . . . . . . . . . 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 41928	. . . . . . . . . . 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 41926	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))))) ≠ 𝑉)
44	42, 43	eqnetrrd 3001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))) ≠ 𝑉)
45	9, 11, 22	dvhlmod 41567	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
47	1, 17, 18, 19, 20, 46, 38	lkr0f2 39618	. . . . . . . . . . 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 2977	. . . . . . . . . 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 4730	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ∈ (𝐶 ∖ {𝑄}))
52	25, 51	eqeltrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) ∈ (𝐶 ∖ {𝑄}))
53		eleq1 2825	. . . . . . 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 3139	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
56		eldifsn 4730	. . . . . . . 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 41948	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ 𝐶 ↔ (𝑔 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑔))) = (𝐿‘𝑔)))
60	59	simplbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ 𝐶 → 𝑔 ∈ 𝐹)
61	60	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ 𝐹)
62	1, 17, 18, 19, 20, 58, 61	lkr0f2 39618	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → ((𝐿‘𝑔) = 𝑉 ↔ 𝑔 = 𝑄))
63	62	necon3bid 2977	. . . . . . . . . . . 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 2938	. . . . . . . . 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 41957	. . . . . . . . . . . 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 865	. . . . . . . . . 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 1118	. . . . . . . . 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 1466	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
75	56, 74	sylan2b 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) → ∃𝑧 ∈ (𝑉 ∖ { 0 })𝑔 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
76		eqcom 2744	. . . . . . . . 9 ⊢ ((𝐽‘𝑧) = 𝑔 ↔ 𝑔 = (𝐽‘𝑧))
77	22	ad2antrr 727	. . . . . . . . . . 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 42006	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
80	79	eqeq2d 2748	. . . . . . . . 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 3160	. . . . . . 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 2735	. 2 ⊢ (𝜑 → ran 𝐽 = (𝐶 ∖ {𝑄}))
88	22	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑡 ∈ (𝑉 ∖ { 0 }) ∧ 𝑢 ∈ (𝑉 ∖ { 0 }))) ∧ (𝐽‘𝑡) = (𝐽‘𝑢)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
89		eqid 2737	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90		eqid 2737	. . . . 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 42006	. . . . . 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 42006	. . . . . 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 2780	. . . . 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 41956	. . . 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 3181	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
100		dff1o6 7221	. 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 1345	1 ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887  {csn 4568   ↦ cmpt 5167  ran crn 5623   Fn wfn 6485  –1-1-onto→wf1o 6489  ‘cfv 6490  ℩crio 7314  (class class class)co 7358  Basecbs 17168  +_gcplusg 17209  Scalarcsca 17212   ·_𝑠 cvsca 17213  0_gc0g 17391  LModclmod 20844  LFnlclfn 39514  LKerclk 39542  LDualcld 39580  HLchlt 39807  LHypclh 40441  DVecHcdvh 41535  ocHcoch 41804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-riotaBAD 39410

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8167  df-undef 8214  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-n0 12427  df-z 12514  df-uz 12778  df-fz 13451  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-0g 17393  df-proset 18249  df-poset 18268  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18387  df-clat 18454  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-submnd 18741  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19088  df-cntz 19281  df-lsm 19600  df-cmn 19746  df-abl 19747  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-oppr 20306  df-dvdsr 20326  df-unit 20327  df-invr 20357  df-dvr 20370  df-drng 20697  df-lmod 20846  df-lss 20916  df-lsp 20956  df-lvec 21088  df-lsatoms 39433  df-lshyp 39434  df-lfl 39515  df-lkr 39543  df-ldual 39581  df-oposet 39633  df-ol 39635  df-oml 39636  df-covers 39723  df-ats 39724  df-atl 39755  df-cvlat 39779  df-hlat 39808  df-llines 39955  df-lplanes 39956  df-lvols 39957  df-lines 39958  df-psubsp 39960  df-pmap 39961  df-padd 40253  df-lhyp 40445  df-laut 40446  df-ldil 40561  df-ltrn 40562  df-trl 40616  df-tgrp 41200  df-tendo 41212  df-edring 41214  df-dveca 41460  df-disoa 41486  df-dvech 41536  df-dib 41596  df-dic 41630  df-dih 41686  df-doch 41805  df-djh 41852

This theorem is referenced by:  lcf1o  42008