lcfrlem16 - Mathbox for Norm Megill

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Theorem lcfrlem16 41582

Description: Lemma for lcfr 41609. (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 3943	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
3		lcfrlem16.m	. . . 4 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
4	2, 3	eleqtrdi 2845	. . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
5		eliun 4976	. . 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 2736	. . . . 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 41133	. . . . . 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 2736	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		lcfrlem16.p	. . . . . . . . 9 ⊢ 𝑃 = (LSubSp‘𝐷)
21	19, 20	lssel 20899	. . . . . . . 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 41134	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	9, 11, 19, 23	ldualvbase 39149	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = 𝐹)
26	22, 25	eleqtrd 2837	. . . . . 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 39119	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ 𝑉)
39	13, 14, 29, 28	dochssv 41379	. . . . . . . . . . . . 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 3133	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
42		iunss 5026	. . . . . . . . . . 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 4002	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
45	44	ssdifd 4125	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∖ { 0 }) ⊆ (𝑉 ∖ { 0 }))
46	45, 1	sseldd 3964	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
47	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem10 41576	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
48	47	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐽‘𝑋) ∈ 𝐹)
49		eqid 2736	. . . . . . 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 4771	. . . . . . . . . . 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 4767	. . . . . . . . 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 41494	. . . . . . 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 41581	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
59		eldifsn 4767	. . . . . . . . . 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 41494	. . . . . . . 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 39030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
65		eqid 2736	. . . . . . 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 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐶)
70	13, 65, 28, 14, 9, 10, 34, 50, 27	lcfl5 41520	. . . . . . . 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 41579	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (𝐶 ∖ {𝑄}))
73	72	eldifad 3943	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐶)
74	13, 65, 28, 14, 9, 10, 34, 15, 47	lcfl5 41520	. . . . . . . . 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 41397	. . . . . 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 39188	. . . 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 39181	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → (𝑘( ·_𝑠 ‘𝐷)𝑔) ∈ 𝐺)
87		eleq1 2823	. . . . . 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 3142	. . . 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 3146	. 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 2933 ∀wral 3052 ∃wrex 3061 {crab 3420 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 ∪ ciun 4972 ↦ cmpt 5206 ran crn 5660 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 Basecbs 17233 +_gcplusg 17276 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 LModclmod 20822 LSubSpclss 20893 LVecclvec 21065 LSAtomsclsa 38997 LFnlclfn 39080 LKerclk 39108 LDualcld 39146 HLchlt 39373 LHypclh 40008 DVecHcdvh 41102 DIsoHcdih 41252 ocHcoch 41371

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-riotaBAD 38976

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-0g 17460 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-lsm 19622 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-lvec 21066 df-lsatoms 38999 df-lshyp 39000 df-lfl 39081 df-lkr 39109 df-ldual 39147 df-oposet 39199 df-ol 39201 df-oml 39202 df-covers 39289 df-ats 39290 df-atl 39321 df-cvlat 39345 df-hlat 39374 df-llines 39522 df-lplanes 39523 df-lvols 39524 df-lines 39525 df-psubsp 39527 df-pmap 39528 df-padd 39820 df-lhyp 40012 df-laut 40013 df-ldil 40128 df-ltrn 40129 df-trl 40183 df-tgrp 40767 df-tendo 40779 df-edring 40781 df-dveca 41027 df-disoa 41053 df-dvech 41103 df-dib 41163 df-dic 41197 df-dih 41253 df-doch 41372 df-djh 41419

This theorem is referenced by: lcfrlem27 41593 lcfrlem37 41603

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