lshpkrlem5 - Mathbox for Norm Megill

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Theorem lshpkrlem5 35800

Description: Lemma for lshpkrex 35804. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.)

**Hypotheses**
Ref	Expression
lshpkrlem.v	⊢ 𝑉 = (Base‘𝑊)
lshpkrlem.a	⊢ + = (+_g‘𝑊)
lshpkrlem.n	⊢ 𝑁 = (LSpan‘𝑊)
lshpkrlem.p	⊢ ⊕ = (LSSum‘𝑊)
lshpkrlem.h	⊢ 𝐻 = (LSHyp‘𝑊)
lshpkrlem.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lshpkrlem.u	⊢ (𝜑 → 𝑈 ∈ 𝐻)
lshpkrlem.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lshpkrlem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lshpkrlem.e	⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)
lshpkrlem.d	⊢ 𝐷 = (Scalar‘𝑊)
lshpkrlem.k	⊢ 𝐾 = (Base‘𝐷)
lshpkrlem.t	⊢ · = ( ·_𝑠 ‘𝑊)
lshpkrlem.o	⊢ 0 = (0_g‘𝐷)
lshpkrlem.g	⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))

**Assertion**
Ref	Expression
lshpkrlem5	⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)))

Distinct variable groups: 𝑥,𝑘,𝑦, + 𝑘,𝐾,𝑥 0 ,𝑘 · ,𝑘,𝑥,𝑦 𝑈,𝑘,𝑥,𝑦 𝑥,𝑉 𝑘,𝑋,𝑥,𝑦 𝑘,𝑍,𝑥,𝑦 𝑧,𝑙, + 𝐺,𝑙,𝑧 𝐾,𝑙 𝑈,𝑙,𝑧 𝑋,𝑙,𝑧 𝑍,𝑙,𝑧,𝑘,𝑥,𝑦 · ,𝑙,𝑧 𝑢,𝑘,𝑣,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) 𝐷(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) + (𝑣,𝑢,𝑠,𝑟) ⊕ (𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) · (𝑣,𝑢,𝑠,𝑟) 𝑈(𝑣,𝑢,𝑠,𝑟) 𝐺(𝑥,𝑦,𝑣,𝑢,𝑘,𝑠,𝑟) 𝐻(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) 𝐾(𝑦,𝑧,𝑣,𝑢,𝑠,𝑟) 𝑁(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) 𝑉(𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) 𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘,𝑠,𝑟,𝑙) 𝑋(𝑣,𝑢,𝑠,𝑟) 0 (𝑥,𝑦,𝑧,𝑣,𝑢,𝑠,𝑟,𝑙) 𝑍(𝑣,𝑢,𝑠,𝑟)

**Proof of Theorem lshpkrlem5**
Step	Hyp	Ref	Expression
1		lshpkrlem.a	. . 3 ⊢ + = (+_g‘𝑊)
2		eqid 2795	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
3		eqid 2795	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
4		simp11 1196	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝜑)
5		lshpkrlem.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	4, 5	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑊 ∈ LVec)
7		lveclmod 19568	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
8	6, 7	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑊 ∈ LMod)
9		eqid 2795	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
10	9	lsssssubg 19420	. . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
11	8, 10	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
12		lshpkrlem.h	. . . . . 6 ⊢ 𝐻 = (LSHyp‘𝑊)
13	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
14		lshpkrlem.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐻)
15	9, 12, 13, 14	lshplss 35667	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊))
16	4, 15	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑈 ∈ (LSubSp‘𝑊))
17	11, 16	sseldd 3890	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑈 ∈ (SubGrp‘𝑊))
18		lshpkrlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
19	4, 18	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑍 ∈ 𝑉)
20		lshpkrlem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
21		lshpkrlem.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
22	20, 9, 21	lspsncl 19439	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
23	8, 19, 22	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
24	11, 23	sseldd 3890	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
25		lshpkrlem.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝑊)
26		lshpkrlem.e	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)
27	20, 2, 21, 25, 12, 5, 14, 18, 26	lshpdisj 35673	. . . 4 ⊢ (𝜑 → (𝑈 ∩ (𝑁‘{𝑍})) = {(0_g‘𝑊)})
28	4, 27	syl 17	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑈 ∩ (𝑁‘{𝑍})) = {(0_g‘𝑊)})
29		lmodabl 19371	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
30	8, 29	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑊 ∈ Abel)
31	3, 30, 17, 24	ablcntzd 18700	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑈 ⊆ ((Cntz‘𝑊)‘(𝑁‘{𝑍})))
32		simp23r 1288	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑧 ∈ 𝑈)
33		simp12 1197	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑙 ∈ 𝐾)
34		simp22 1200	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑟 ∈ 𝑈)
35		lshpkrlem.d	. . . . . 6 ⊢ 𝐷 = (Scalar‘𝑊)
36		lshpkrlem.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
37		lshpkrlem.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐷)
38	35, 36, 37, 9	lssvscl 19417	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ 𝐾 ∧ 𝑟 ∈ 𝑈)) → (𝑙 · 𝑟) ∈ 𝑈)
39	8, 16, 33, 34, 38	syl22anc 835	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑙 · 𝑟) ∈ 𝑈)
40		simp23l 1287	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑠 ∈ 𝑈)
41	1, 9	lssvacl 19416	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ ((𝑙 · 𝑟) ∈ 𝑈 ∧ 𝑠 ∈ 𝑈)) → ((𝑙 · 𝑟) + 𝑠) ∈ 𝑈)
42	8, 16, 39, 40, 41	syl22anc 835	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝑙 · 𝑟) + 𝑠) ∈ 𝑈)
43		simp13 1198	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑢 ∈ 𝑉)
44	20, 35, 36, 37	lmodvscl 19341	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) → (𝑙 · 𝑢) ∈ 𝑉)
45	8, 33, 43, 44	syl3anc 1364	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑙 · 𝑢) ∈ 𝑉)
46		simp21 1199	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑣 ∈ 𝑉)
47	20, 1	lmodvacl 19338	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑙 · 𝑢) ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) → ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉)
48	8, 45, 46, 47	syl3anc 1364	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉)
49	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → 𝑊 ∈ LVec)
50	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → 𝑈 ∈ 𝐻)
51	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → 𝑍 ∈ 𝑉)
52		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉)
53	26	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)
54		lshpkrlem.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
55		lshpkrlem.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))
56	20, 1, 21, 25, 12, 49, 50, 51, 52, 53, 35, 37, 36, 54, 55	lshpkrlem2 35797	. . . . 5 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) ∈ 𝐾)
57	4, 48, 56	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) ∈ 𝐾)
58	20, 36, 35, 37, 21, 8, 57, 19	lspsneli 19463	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍) ∈ (𝑁‘{𝑍}))
59	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑊 ∈ LVec)
60	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑈 ∈ 𝐻)
61	18	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑍 ∈ 𝑉)
62		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑢 ∈ 𝑉)
63	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)
64	20, 1, 21, 25, 12, 59, 60, 61, 62, 63, 35, 37, 36, 54, 55	lshpkrlem2 35797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (𝐺‘𝑢) ∈ 𝐾)
65	4, 43, 64	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘𝑢) ∈ 𝐾)
66		eqid 2795	. . . . . . 7 ⊢ (._r‘𝐷) = (._r‘𝐷)
67	35, 37, 66	lmodmcl 19336	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ (𝐺‘𝑢) ∈ 𝐾) → (𝑙(._r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾)
68	8, 33, 65, 67	syl3anc 1364	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑙(._r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾)
69	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec)
70	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ 𝐻)
71	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑍 ∈ 𝑉)
72		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
73	26	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)
74	20, 1, 21, 25, 12, 69, 70, 71, 72, 73, 35, 37, 36, 54, 55	lshpkrlem2 35797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾)
75	4, 46, 74	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘𝑣) ∈ 𝐾)
76		eqid 2795	. . . . . 6 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
77	35, 37, 76	lmodacl 19335	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑙(._r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾 ∧ (𝐺‘𝑣) ∈ 𝐾) → ((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) ∈ 𝐾)
78	8, 68, 75, 77	syl3anc 1364	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) ∈ 𝐾)
79	20, 36, 35, 37, 21, 8, 78, 19	lspsneli 19463	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍) ∈ (𝑁‘{𝑍}))
80		simp33 1204	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))
81		simp1 1129	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉))
82	20, 9	lssel 19399	. . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑉)
83	16, 34, 82	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑟 ∈ 𝑉)
84	20, 9	lssel 19399	. . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝑉)
85	16, 40, 84	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑠 ∈ 𝑉)
86		simp31 1202	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)))
87		simp32 1203	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))
88		lshpkrlem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
89	20, 1, 21, 25, 12, 5, 14, 18, 88, 26, 35, 37, 36, 54, 55	lshpkrlem4 35799	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍)))
90	81, 46, 83, 85, 86, 87, 89	syl132anc 1381	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍)))
91	80, 90	eqtr3d 2833	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍)))
92	1, 2, 3, 17, 24, 28, 31, 32, 42, 58, 79, 91	subgdisj2 18545	. 2 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍) = (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍))
93	20, 21, 25, 12, 2, 13, 14, 18, 26	lshpne0 35672	. . . 4 ⊢ (𝜑 → 𝑍 ≠ (0_g‘𝑊))
94	4, 93	syl 17	. . 3 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → 𝑍 ≠ (0_g‘𝑊))
95	20, 36, 35, 37, 2, 6, 57, 78, 19, 94	lvecvscan2 19574	. 2 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍) = (((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)) · 𝑍) ↔ (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣))))
96	92, 95	mpbid 233	1 ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(._r‘𝐷)(𝐺‘𝑢))(+_g‘𝐷)(𝐺‘𝑣)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃wrex 3106 ∩ cin 3858 ⊆ wss 3859 {csn 4472 ↦ cmpt 5041 ‘cfv 6225 ℩crio 6976 (class class class)co 7016 Basecbs 16312 +_gcplusg 16394 ._rcmulr 16395 Scalarcsca 16397 ·_𝑠 cvsca 16398 0_gc0g 16542 SubGrpcsubg 18027 Cntzccntz 18186 LSSumclsm 18489 Abelcabl 18634 LModclmod 19324 LSubSpclss 19393 LSpanclspn 19433 LVecclvec 19564 LSHypclsh 35661

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lshyp 35663

This theorem is referenced by: lshpkrlem6 35801

W3C validator