lcfrlem16 - Mathbox for Norm Megill

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Theorem lcfrlem16 41161

Description: Lemma for lcfr 41188. (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 3956	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
3		lcfrlem16.m	. . . 4 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
4	2, 3	eleqtrdi 2835	. . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
5		eliun 5001	. . 3 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
6	4, 5	sylib 217	. 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 2725	. . . . 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 40712	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
17	16	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LVec)
18		lcfrlem16.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝑃)
19		eqid 2725	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		lcfrlem16.p	. . . . . . . . 9 ⊢ 𝑃 = (LSubSp‘𝐷)
21	19, 20	lssel 20833	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑃 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
22	18, 21	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
23	13, 14, 15	dvhlmod 40713	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	9, 11, 19, 23	ldualvbase 38728	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
25	24	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = 𝐹)
26	22, 25	eleqtrd 2827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ 𝐹)
27	26	3adant3 1129	. . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	23	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod)
38	29, 9, 10, 37, 26	lkrssv 38698	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ 𝑉)
39	13, 14, 29, 28	dochssv 40958	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
40	36, 38, 39	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
41	40	ralrimiva 3135	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
42		iunss 5049	. . . . . . . . . . 11 ⊢ (∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉 ↔ ∀𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
43	41, 42	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑉)
44	3, 43	eqsstrid 4025	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
45	44	ssdifd 4137	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∖ { 0 }) ⊆ (𝑉 ∖ { 0 }))
46	45, 1	sseldd 3977	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
47	13, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46	lcfrlem10 41155	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
48	47	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐽‘𝑋) ∈ 𝐹)
49		eqid 2725	. . . . . . 7 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
50	15	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51		simp3 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
52		eldifsni 4795	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) → 𝑋 ≠ 0 )
53	1, 52	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 0 )
54	53	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ≠ 0 )
55		eldifsn 4792	. . . . . . . . 9 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝑔)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑋 ≠ 0 ))
56	51, 54, 55	sylanbrc 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝑔)) ∖ { 0 }))
57	13, 28, 14, 29, 32, 9, 10, 50, 27, 56, 49	dochsnkrlem2 41073	. . . . . . 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 41160	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
59		eldifsn 4792	. . . . . . . . . 10 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∧ 𝑋 ≠ 0 ))
60	58, 53, 59	sylanbrc 581	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∖ { 0 }))
61	13, 28, 14, 29, 32, 9, 10, 15, 47, 60, 49	dochsnkrlem2 41073	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∈ (LSAtoms‘𝑈))
62	61	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ∈ (LSAtoms‘𝑈))
63	58	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
64	32, 49, 17, 57, 62, 54, 51, 63	lsat2el 38609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
65		eqid 2725	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
66		lcfrlem16.gs	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
67	66	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝐺 ⊆ 𝐶)
68		simp2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐺)
69	67, 68	sseldd 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑔 ∈ 𝐶)
70	13, 65, 28, 14, 9, 10, 34, 50, 27	lcfl5 41099	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑔 ∈ 𝐶 ↔ (𝐿‘𝑔) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
71	69, 70	mpbid 231	. . . . . . 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 41158	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (𝐶 ∖ {𝑄}))
73	72	eldifad 3956	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐶)
74	13, 65, 28, 14, 9, 10, 34, 15, 47	lcfl5 41099	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽‘𝑋) ∈ 𝐶 ↔ (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
75	73, 74	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	75	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐿‘(𝐽‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	13, 65, 28, 50, 71, 76	doch11 40976	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) ↔ (𝐿‘𝑔) = (𝐿‘(𝐽‘𝑋))))
78	64, 77	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝐿‘𝑔) = (𝐿‘(𝐽‘𝑋)))
79	7, 8, 9, 10, 11, 12, 17, 27, 48, 78	eqlkr4 38767	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ∃𝑘 ∈ 𝑅 (𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔))
80	23	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod)
81	80	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑈 ∈ LMod)
82	18	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝐺 ∈ 𝑃)
83	82	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝐺 ∈ 𝑃)
84		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑘 ∈ 𝑅)
85		simpl2 1189	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → 𝑔 ∈ 𝐺)
86	7, 8, 11, 12, 20, 81, 83, 84, 85	ldualssvscl 38760	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → (𝑘( ·_𝑠 ‘𝐷)𝑔) ∈ 𝐺)
87		eleq1 2813	. . . . . 6 ⊢ ((𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔) → ((𝐽‘𝑋) ∈ 𝐺 ↔ (𝑘( ·_𝑠 ‘𝐷)𝑔) ∈ 𝐺))
88	86, 87	syl5ibrcom 246	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺 ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) ∧ 𝑘 ∈ 𝑅) → ((𝐽‘𝑋) = (𝑘( ·_𝑠 ‘𝐷)𝑔) → (𝐽‘𝑋) ∈ 𝐺))
89	88	rexlimdva 3144	. . . 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 3148	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 {crab 3418 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 ∪ ciun 4997 ↦ cmpt 5232 ran crn 5679 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17183 +_gcplusg 17236 Scalarcsca 17239 ·_𝑠 cvsca 17240 0_gc0g 17424 LModclmod 20755 LSubSpclss 20827 LVecclvec 20999 LSAtomsclsa 38576 LFnlclfn 38659 LKerclk 38687 LDualcld 38725 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 DIsoHcdih 40831 ocHcoch 40950

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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998

This theorem is referenced by: lcfrlem27 41172 lcfrlem37 41182

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