Theorem lcfrlem9 40924

Description: Lemma for lcf1o 40925. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 40925.) 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 6896	. . . . 5 ⊢ 𝑉 ∈ V
3	2	mptex 7217	. . . 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 6684	. . 3 ⊢ 𝐽 Fn (𝑉 ∖ { 0 })
6	5	a1i 11	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑉 ∖ { 0 }))
7		fvelrnb 6943	. . . . 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 40923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
26		eqid 2724	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
27		sneq 4631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
28	27	fveq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ( ⊥ ‘{𝑦}) = ( ⊥ ‘{𝑧}))
29		oveq2 7410	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑘 · 𝑦) = (𝑘 · 𝑧))
30	29	oveq2d 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑤 + (𝑘 · 𝑦)) = (𝑤 + (𝑘 · 𝑧)))
31	30	eqeq2d 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑧))))
32	28, 31	rexeqbidv 3335	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
33	32	riotabidv 7360	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑦})𝑣 = (𝑤 + (𝑘 · 𝑦))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧))))
34	33	mpteq2dv 5241	. . . . . . . . . . . . 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 3626	. . . . . . . . . . . 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 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 40849	. . . . . . . . . . 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 40874	. . . . . . . . . 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 40848	. . . . . . . . . . . 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 40845	. . . . . . . . . . 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 40843	. . . . . . . . . . 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 40484	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → 𝑈 ∈ LMod)
47	1, 17, 18, 19, 20, 46, 38	lkr0f2 38534	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))) ≠ 𝑄)
50		eldifsn 4783	. . . . . . . . 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 2825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) ∈ (𝐶 ∖ {𝑄}))
53		eleq1 2813	. . . . . . 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 3147	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ (𝑉 ∖ { 0 })(𝐽‘𝑧) = 𝑔 → 𝑔 ∈ (𝐶 ∖ {𝑄})))
56		eldifsn 4783	. . . . . . . 8 ⊢ (𝑔 ∈ (𝐶 ∖ {𝑄}) ↔ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄))
57		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐶 ∧ 𝑔 ≠ 𝑄)) → 𝑔 ∈ 𝐶)
58	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑈 ∈ LMod)
59	21	lcfl1lem 40865	. . . . . . . . . . . . . . . 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 38534	. . . . . . . . . . . . 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 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 2937	. . . . . . . . 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 40874	. . . . . . . . . . . 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 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 1114	. . . . . . . . 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 1459	. . . . . . . 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 2731	. . . . . . . . 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 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 40923	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐶 ∖ {𝑄})) ∧ 𝑧 ∈ (𝑉 ∖ { 0 })) → (𝐽‘𝑧) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑧})𝑣 = (𝑤 + (𝑘 · 𝑧)))))
80	79	eqeq2d 2735	. . . . . . . . 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 3168	. . . . . . 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 211	. . . 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 2722	. 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 2724	. . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑡})𝑣 = (𝑤 + (𝑘 · 𝑡))))
90		eqid 2724	. . . . 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 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 40923	. . . . . 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 40923	. . . . . 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 2772	. . . . 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 40873	. . . 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 3192	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑉 ∖ { 0 })∀𝑢 ∈ (𝑉 ∖ { 0 })((𝐽‘𝑡) = (𝐽‘𝑢) → 𝑡 = 𝑢))
100		dff1o6 7266	. 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 1340	1 ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  {crab 3424   ∖ cdif 3938  {csn 4621   ↦ cmpt 5222  ran crn 5668   Fn wfn 6529  –1-1-onto→wf1o 6533  ‘cfv 6534  ℩crio 7357  (class class class)co 7402  Basecbs 17149  +_gcplusg 17202  Scalarcsca 17205   ·_𝑠 cvsca 17206  0_gc0g 17390  LModclmod 20702  LFnlclfn 38430  LKerclk 38458  LDualcld 38496  HLchlt 38723  LHypclh 39358  DVecHcdvh 40452  ocHcoch 40721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-riotaBAD 38326

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-tpos 8207  df-undef 8254  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13486  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-0g 17392  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-clat 18460  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18710  df-grp 18862  df-minusg 18863  df-sbg 18864  df-subg 19046  df-cntz 19229  df-lsm 19552  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136  df-oppr 20232  df-dvdsr 20255  df-unit 20256  df-invr 20286  df-dvr 20299  df-drng 20585  df-lmod 20704  df-lss 20775  df-lsp 20815  df-lvec 20947  df-lsatoms 38349  df-lshyp 38350  df-lfl 38431  df-lkr 38459  df-ldual 38497  df-oposet 38549  df-ol 38551  df-oml 38552  df-covers 38639  df-ats 38640  df-atl 38671  df-cvlat 38695  df-hlat 38724  df-llines 38872  df-lplanes 38873  df-lvols 38874  df-lines 38875  df-psubsp 38877  df-pmap 38878  df-padd 39170  df-lhyp 39362  df-laut 39363  df-ldil 39478  df-ltrn 39479  df-trl 39533  df-tgrp 40117  df-tendo 40129  df-edring 40131  df-dveca 40377  df-disoa 40403  df-dvech 40453  df-dib 40513  df-dic 40547  df-dih 40603  df-doch 40722  df-djh 40769

This theorem is referenced by:  lcf1o  40925