lshpkrlem5 - Mathbox for Norm Megill

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Theorem lshpkrlem5 34401

Description: Lemma for lshpkrex 34405. 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 2622	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
3		eqid 2622	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
4		simp11 1091	. . . . . . 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 19106	. . . . . 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 2622	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
10	9	lsssssubg 18958	. . . . 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 34268	. . . . 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 3604	. . 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 18977	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
23	8, 19, 22	syl2anc 693	. . . 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 3604	. . 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 34274	. . . 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 18910	. . . . 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 18260	. . 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 1183	. . 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 1092	. . . . 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 1095	. . . . 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 18955	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ 𝐾 ∧ 𝑟 ∈ 𝑈)) → (𝑙 · 𝑟) ∈ 𝑈)
39	8, 16, 33, 34, 38	syl22anc 1327	. . . 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 1182	. . . 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 18954	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ ((𝑙 · 𝑟) ∈ 𝑈 ∧ 𝑠 ∈ 𝑈)) → ((𝑙 · 𝑟) + 𝑠) ∈ 𝑈)
42	8, 16, 39, 40, 41	syl22anc 1327	. . 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 1093	. . . . . . 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 18880	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) → (𝑙 · 𝑢) ∈ 𝑉)
45	8, 33, 43, 44	syl3anc 1326	. . . . . 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 1094	. . . . . 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 18877	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑙 · 𝑢) ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) → ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉)
48	8, 45, 46, 47	syl3anc 1326	. . . . 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 477	. . . . . 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 34398	. . . . 5 ⊢ ((𝜑 ∧ ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) ∈ 𝐾)
57	4, 48, 56	syl2anc 693	. . . 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 19001	. . 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 477	. . . . . . . 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 34398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (𝐺‘𝑢) ∈ 𝐾)
65	4, 43, 64	syl2anc 693	. . . . . 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 2622	. . . . . . 7 ⊢ (._r‘𝐷) = (._r‘𝐷)
67	35, 37, 66	lmodmcl 18875	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ (𝐺‘𝑢) ∈ 𝐾) → (𝑙(._r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾)
68	8, 33, 65, 67	syl3anc 1326	. . . . 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 477	. . . . . . 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 34398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾)
75	4, 46, 74	syl2anc 693	. . . . 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 2622	. . . . . 6 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
77	35, 37, 76	lmodacl 18874	. . . . 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 1326	. . . 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 19001	. . 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 1099	. . . 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 1061	. . . . 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 18938	. . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑉)
83	16, 34, 82	syl2anc 693	. . . . 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 18938	. . . . . 6 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝑉)
85	16, 40, 84	syl2anc 693	. . . . 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 1097	. . . . 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 1098	. . . . 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 34400	. . . . 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 1344	. . . 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 2658	. . 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 18105	. 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 34273	. . . 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 19112	. 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 222	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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 ._rcmulr 15942 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 SubGrpcsubg 17588 Cntzccntz 17748 LSSumclsm 18049 Abelcabl 18194 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 LVecclvec 19102 LSHypclsh 34262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lshyp 34264

This theorem is referenced by: lshpkrlem6 34402

W3C validator