lshpkrlem5 - Mathbox for Norm Megill

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Theorem lshpkrlem5 37128

Description: Lemma for lshpkrex 37132. 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 2738	. . 3 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
3		eqid 2738	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
4		simp11 1202	. . . . . . 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 20368	. . . . . 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 2738	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
10	9	lsssssubg 20220	. . . . 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 36995	. . . . 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 3922	. . 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 20239	. . . . 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 3922	. . 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 37001	. . . 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 20170	. . . . 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 19458	. . 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 1294	. . 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 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)))) → 𝑙 ∈ 𝐾)
34		simp22 1206	. . . . 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 20217	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ 𝐾 ∧ 𝑟 ∈ 𝑈)) → (𝑙 · 𝑟) ∈ 𝑈)
39	8, 16, 33, 34, 38	syl22anc 836	. . . 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 1293	. . . 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 20216	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝑊)) ∧ ((𝑙 · 𝑟) ∈ 𝑈 ∧ 𝑠 ∈ 𝑈)) → ((𝑙 · 𝑟) + 𝑠) ∈ 𝑈)
42	8, 16, 39, 40, 41	syl22anc 836	. . 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 1204	. . . . . . 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 20140	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) → (𝑙 · 𝑢) ∈ 𝑉)
45	8, 33, 43, 44	syl3anc 1370	. . . . . 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 1205	. . . . . 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 20137	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑙 · 𝑢) ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) → ((𝑙 · 𝑢) + 𝑣) ∈ 𝑉)
48	8, 45, 46, 47	syl3anc 1370	. . . . 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 37125	. . . . 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 20263	. . 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 37125	. . . . . . 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 2738	. . . . . . 7 ⊢ (._r‘𝐷) = (._r‘𝐷)
67	35, 37, 66	lmodmcl 20135	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ (𝐺‘𝑢) ∈ 𝐾) → (𝑙(._r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾)
68	8, 33, 65, 67	syl3anc 1370	. . . . 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 37125	. . . . . 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 2738	. . . . . 6 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
77	35, 37, 76	lmodacl 20134	. . . . 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 1370	. . . 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 20263	. . 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 1210	. . . 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 1135	. . . . 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 20199	. . . . . 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 20199	. . . . . 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 1208	. . . . 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 1209	. . . . 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 37127	. . . . 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 1387	. . . 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 2780	. . 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 19298	. 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 37000	. . . 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 20374	. 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 231	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 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 {csn 4561 ↦ cmpt 5157 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 SubGrpcsubg 18749 Cntzccntz 18921 LSSumclsm 19239 Abelcabl 19387 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 LVecclvec 20364 LSHypclsh 36989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lshyp 36991

This theorem is referenced by: lshpkrlem6 37129

W3C validator