lcfrlem16 - Mathbox for Norm Megill

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Theorem lcfrlem16 41537

Description: Lemma for lcfr 41564. (Contributed by NM, 8-Mar-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 = (0_g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0_g‘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))
lcfrlem16.p	⊢ 𝑃 = (LSubSp‘𝐷)
lcfrlem16.g	⊢ (𝜑 → 𝐺 ∈ 𝑃)
lcfrlem16.gs	⊢ (𝜑 → 𝐺 ⊆ 𝐶)
lcfrlem16.m	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem16.x	⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 }))

**Assertion**
Ref	Expression
lcfrlem16	⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)

Distinct variable groups: 𝑥,𝑤, ⊥ 𝑥, 0 𝑥,𝑣,𝑉 𝑥, · 𝑣,𝑘,𝑤,𝑥,𝑋 𝑥, + 𝑥,𝑅 𝑓,𝑘,𝑣,𝑤, + 𝑓,𝐹,𝑘 𝑔,𝑘,𝐺 𝑓,𝑔,𝐽,𝑘 𝑓,𝐿,𝑘 ⊥ ,𝑓,𝑘,𝑣 𝑅,𝑓,𝑘,𝑣 𝑆,𝑘 · ,𝑓,𝑘,𝑣,𝑤 𝑈,𝑘 𝑓,𝑉,𝑔,𝑥 𝑓,𝑋 𝑣,𝑔,𝑤,𝑥,𝑋 𝜑,𝑔,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑓) 𝐶(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑃(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) + (𝑔) 𝑄(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑅(𝑤,𝑔) 𝑆(𝑥,𝑤,𝑣,𝑓,𝑔) · (𝑔) 𝑈(𝑥,𝑤,𝑣,𝑓,𝑔) 𝐸(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑔) 𝐺(𝑥,𝑤,𝑣,𝑓) 𝐻(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐽(𝑥,𝑤,𝑣) 𝐾(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐿(𝑥,𝑤,𝑣,𝑔) ⊥ (𝑔) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 0 (𝑤,𝑣,𝑓,𝑔,𝑘)

**Proof of Theorem lcfrlem16**
Step	Hyp	Ref	Expression
1		lcfrlem16.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 }))
2	1	eldifad 3917	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
3		lcfrlem16.m	. . . 4 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
4	2, 3	eleqtrdi 2838	. . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
5		eliun 4948	. . 3 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
7		lcf1o.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑈)
8		lcf1o.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
9		lcf1o.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑈)
10		lcf1o.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
11		lcf1o.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
12		eqid 2729	. . . . 5 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
13		lcf1o.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
14		lcf1o.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
15		lcflo.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16	13, 14, 15	dvhlvec 41088	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
17	16	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LVec)
18		lcfrlem16.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝑃)
19		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		lcfrlem16.p	. . . . . . . . 9 ⊢ 𝑃 = (LSubSp‘𝐷)
21	19, 20	lssel 20858	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑃 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
22	18, 21	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
23	13, 14, 15	dvhlmod 41089	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	9, 11, 19, 23	ldualvbase 39104	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = 𝐹)
26	22, 25	eleqtrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ 𝐹)
27	26	3adant3 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐹)
28		lcf1o.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
29		lcf1o.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
30		lcf1o.a	. . . . . . 7 ⊢ + = (+_g‘𝑈)
31		lcf1o.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
32		lcf1o.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
33		lcf1o.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐷)
34		lcf1o.c	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
35		lcf1o.j	. . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
36	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	23	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod)
38	29, 9, 10, 37, 26	lkrssv 39074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ 𝑉)
39	13, 14, 29, 28	dochssv 41334	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
40	36, 38, 39	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
41	40	ralrimiva 3121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
42		iunss 4997	. . . . . . . . . . 11 ⊢ (∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉 ↔ ∀𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
43	41, 42	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
44	3, 43	eqsstrid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
45	44	ssdifd 4098	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∖ { 0 }) ⊆ (𝑉 ∖ { 0 }))
46	45, 1	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
47	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem10 41531	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
48	47	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐽‘𝑋) ∈ 𝐹)
49		eqid 2729	. . . . . . 7 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
50	15	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
52		eldifsni 4744	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) → 𝑋 ≠ 0 )
53	1, 52	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 0 )
54	53	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ≠ 0 )
55		eldifsn 4740	. . . . . . . . 9 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝑔)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑋 ≠ 0 ))
56	51, 54, 55	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝑔)) ∖ { 0 }))
57	13, 28, 14, 29, 32, 9, 10, 50, 27, 56, 49	dochsnkrlem2 41449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSAtoms‘𝑈))
58	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem15 41536	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
59		eldifsn 4740	. . . . . . . . . 10 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∧ 𝑋 ≠ 0 ))
60	58, 53, 59	sylanbrc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∖ { 0 }))
61	13, 28, 14, 29, 32, 9, 10, 15, 47, 60, 49	dochsnkrlem2 41449	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∈ (LSAtoms‘𝑈))
62	61	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∈ (LSAtoms‘𝑈))
63	58	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
64	32, 49, 17, 57, 62, 54, 51, 63	lsat2el 38985	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
65		eqid 2729	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
66		lcfrlem16.gs	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
67	66	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝐺 ⊆ 𝐶)
68		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐺)
69	67, 68	sseldd 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐶)
70	13, 65, 28, 14, 9, 10, 34, 50, 27	lcfl5 41475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑔 ∈ 𝐶 ↔ (𝐿‘𝑔) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
71	69, 70	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐿‘𝑔) ∈ ran ((DIsoH‘𝐾)‘𝑊))
72	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem13 41534	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (𝐶 ∖ {𝑄}))
73	72	eldifad 3917	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐶)
74	13, 65, 28, 14, 9, 10, 34, 15, 47	lcfl5 41475	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽‘𝑋) ∈ 𝐶 ↔ (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
75	73, 74	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	75	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	13, 65, 28, 50, 71, 76	doch11 41352	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ↔ (𝐿‘𝑔) = (𝐿‘(𝐽‘𝑋))))
78	64, 77	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐿‘𝑔) = (𝐿‘(𝐽‘𝑋)))
79	7, 8, 9, 10, 11, 12, 17, 27, 48, 78	eqlkr4 39143	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ∃𝑘 ∈ 𝑅 (𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔))
80	23	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod)
81	80	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑈 ∈ LMod)
82	18	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝐺 ∈ 𝑃)
83	82	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝐺 ∈ 𝑃)
84		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑘 ∈ 𝑅)
85		simpl2 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑔 ∈ 𝐺)
86	7, 8, 11, 12, 20, 81, 83, 84, 85	ldualssvscl 39136	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → (𝑘( ·_𝑠 ‘𝐷)𝑔) ∈ 𝐺)
87		eleq1 2816	. . . . . 6 ⊢ ((𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔) → ((𝐽‘𝑋) ∈ 𝐺 ↔ (𝑘( ·_𝑠 ‘𝐷)𝑔) ∈ 𝐺))
88	86, 87	syl5ibrcom 247	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → ((𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔) → (𝐽‘𝑋) ∈ 𝐺))
89	88	rexlimdva 3130	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (∃𝑘 ∈ 𝑅 (𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔) → (𝐽‘𝑋) ∈ 𝐺))
90	79, 89	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐽‘𝑋) ∈ 𝐺)
91	90	rexlimdv3a 3134	. 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝐽‘𝑋) ∈ 𝐺))
92	6, 91	mpd 15	1 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3396 ∖ cdif 3902 ⊆ wss 3905 {csn 4579 ∪ ciun 4944 ↦ cmpt 5176 ran crn 5624 ‘cfv 6486 ℩crio 7309 (class class class)co 7353 Basecbs 17138 +_gcplusg 17179 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 LModclmod 20781 LSubSpclss 20852 LVecclvec 21024 LSAtomsclsa 38952 LFnlclfn 39035 LKerclk 39063 LDualcld 39101 HLchlt 39328 LHypclh 39963 DVecHcdvh 41057 DIsoHcdih 41207 ocHcoch 41326

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 38931

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17363 df-proset 18218 df-poset 18237 df-plt 18252 df-lub 18268 df-glb 18269 df-join 18270 df-meet 18271 df-p0 18347 df-p1 18348 df-lat 18356 df-clat 18423 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lvec 21025 df-lsatoms 38954 df-lshyp 38955 df-lfl 39036 df-lkr 39064 df-ldual 39102 df-oposet 39154 df-ol 39156 df-oml 39157 df-covers 39244 df-ats 39245 df-atl 39276 df-cvlat 39300 df-hlat 39329 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374

This theorem is referenced by: lcfrlem27 41548 lcfrlem37 41558

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