Theorem imaslmod 33428

Description: The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.)

Hypotheses

Ref	Expression
imaslmod.u	⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀))
imaslmod.v	⊢ 𝑉 = (Base‘𝑀)
imaslmod.k	⊢ 𝑆 = (Base‘(Scalar‘𝑀))
imaslmod.p	⊢ + = (+g‘𝑀)
imaslmod.t	⊢ · = ( ·𝑠 ‘𝑀)
imaslmod.o	⊢ 0 = (0g‘𝑀)
imaslmod.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imaslmod.e1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imaslmod.e2	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏))))
imaslmod.l	⊢ (𝜑 → 𝑀 ∈ LMod)

Assertion

Ref	Expression
imaslmod	⊢ (𝜑 → 𝑁 ∈ LMod)

Distinct variable groups:   𝐵,𝑏,𝑘,𝑝,𝑞   𝐹,𝑎,𝑏,𝑘,𝑝,𝑞   + ,𝑏,𝑘,𝑝,𝑞   𝑀,𝑏,𝑘,𝑝,𝑞   𝑁,𝑎,𝑏,𝑘,𝑝,𝑞   0 ,𝑝,𝑞   𝑆,𝑎,𝑏,𝑘   𝑉,𝑎,𝑏,𝑘,𝑝,𝑞   · ,𝑏,𝑘,𝑝,𝑞   𝜑,𝑎,𝑏,𝑘,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑎)   + (𝑎)   𝑆(𝑞,𝑝)   · (𝑎)   𝑀(𝑎)   0 (𝑘,𝑎,𝑏)

Proof of Theorem imaslmod

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaslmod.u	. . 3 ⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀))
2		imaslmod.v	. . . 4 ⊢ 𝑉 = (Base‘𝑀)
3	2	a1i 11	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑀))
4		imaslmod.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
5		imaslmod.l	. . 3 ⊢ (𝜑 → 𝑀 ∈ LMod)
6	1, 3, 4, 5	imasbas 17467	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑁))
7		eqidd 2738	. 2 ⊢ (𝜑 → (+g‘𝑁) = (+g‘𝑁))
8		eqid 2737	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
9	1, 3, 4, 5, 8	imassca 17474	. 2 ⊢ (𝜑 → (Scalar‘𝑀) = (Scalar‘𝑁))
10		eqidd 2738	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁))
11		imaslmod.k	. . 3 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
12	11	a1i 11	. 2 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑀)))
13		eqidd 2738	. 2 ⊢ (𝜑 → (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀)))
14		eqidd 2738	. 2 ⊢ (𝜑 → (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀)))
15		eqidd 2738	. 2 ⊢ (𝜑 → (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
16	8	lmodring 20854	. . 3 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
17	5, 16	syl 17	. 2 ⊢ (𝜑 → (Scalar‘𝑀) ∈ Ring)
18		imaslmod.p	. . . . 5 ⊢ + = (+g‘𝑀)
19	18	a1i 11	. . . 4 ⊢ (𝜑 → + = (+g‘𝑀))
20		imaslmod.e1	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
21		lmodgrp 20853	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
22	5, 21	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Grp)
23		imaslmod.o	. . . 4 ⊢ 0 = (0g‘𝑀)
24	1, 3, 19, 4, 20, 22, 23	imasgrp 19023	. . 3 ⊢ (𝜑 → (𝑁 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑁)))
25	24	simpld 494	. 2 ⊢ (𝜑 → 𝑁 ∈ Grp)
26		imaslmod.t	. . . 4 ⊢ · = ( ·𝑠 ‘𝑀)
27		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
28		imaslmod.e2	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏))))
29	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑀 ∈ LMod)
30		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑘 ∈ 𝑆)
31		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
32	2, 8, 26, 11	lmodvscl 20864	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉) → (𝑘 · 𝑏) ∈ 𝑉)
33	29, 30, 31, 32	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → (𝑘 · 𝑏) ∈ 𝑉)
34	1, 3, 4, 5, 8, 11, 26, 27, 28, 33	imasvscaf 17494	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑁):(𝑆 × 𝐵)⟶𝐵)
35	34	fovcld 7487	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵) → (𝑢( ·𝑠 ‘𝑁)𝑣) ∈ 𝐵)
36		simp-5l 785	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝜑)
37		simpllr 776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
38	37	simp1d 1143	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑢 ∈ 𝑆)
39	38	ad2antrr 727	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑢 ∈ 𝑆)
40	36, 22	syl 17	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑀 ∈ Grp)
41		simplr 769	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑦 ∈ 𝑉)
42		simp-4r 784	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑧 ∈ 𝑉)
43	2, 18	grpcl 18908	. . . . . . . 8 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 + 𝑧) ∈ 𝑉)
44	40, 41, 42, 43	syl3anc 1374	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑦 + 𝑧) ∈ 𝑉)
45	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ (𝑦 + 𝑧) ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑢 · (𝑦 + 𝑧))))
46	36, 39, 44, 45	syl3anc 1374	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑢 · (𝑦 + 𝑧))))
47		eqid 2737	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
48	4, 20, 1, 3, 5, 18, 47	imasaddval 17487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
49	36, 41, 42, 48	syl3anc 1374	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → ((𝐹‘𝑦)(+g‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
50		simpr 484	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘𝑦) = 𝑣)
51		simpllr 776	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘𝑧) = 𝑤)
52	50, 51	oveq12d 7378	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → ((𝐹‘𝑦)(+g‘𝑁)(𝐹‘𝑧)) = (𝑣(+g‘𝑁)𝑤))
53	49, 52	eqtr3d 2774	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘(𝑦 + 𝑧)) = (𝑣(+g‘𝑁)𝑤))
54	53	oveq2d 7376	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑦 + 𝑧))) = (𝑢( ·𝑠 ‘𝑁)(𝑣(+g‘𝑁)𝑤)))
55	36, 5	syl 17	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑀 ∈ LMod)
56	2, 18, 8, 26, 11	lmodvsdi 20871	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑢 · (𝑦 + 𝑧)) = ((𝑢 · 𝑦) + (𝑢 · 𝑧)))
57	55, 39, 41, 42, 56	syl13anc 1375	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢 · (𝑦 + 𝑧)) = ((𝑢 · 𝑦) + (𝑢 · 𝑧)))
58	57	fveq2d 6838	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘(𝑢 · (𝑦 + 𝑧))) = (𝐹‘((𝑢 · 𝑦) + (𝑢 · 𝑧))))
59	46, 54, 58	3eqtr3d 2780	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝑣(+g‘𝑁)𝑤)) = (𝐹‘((𝑢 · 𝑦) + (𝑢 · 𝑧))))
60	2, 8, 26, 11	lmodvscl 20864	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉) → (𝑢 · 𝑦) ∈ 𝑉)
61	55, 39, 41, 60	syl3anc 1374	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢 · 𝑦) ∈ 𝑉)
62	2, 8, 26, 11	lmodvscl 20864	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑢 · 𝑧) ∈ 𝑉)
63	55, 39, 42, 62	syl3anc 1374	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢 · 𝑧) ∈ 𝑉)
64	4, 20, 1, 3, 5, 18, 47	imasaddval 17487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 · 𝑦) ∈ 𝑉 ∧ (𝑢 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑢 · 𝑦))(+g‘𝑁)(𝐹‘(𝑢 · 𝑧))) = (𝐹‘((𝑢 · 𝑦) + (𝑢 · 𝑧))))
65	36, 61, 63, 64	syl3anc 1374	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → ((𝐹‘(𝑢 · 𝑦))(+g‘𝑁)(𝐹‘(𝑢 · 𝑧))) = (𝐹‘((𝑢 · 𝑦) + (𝑢 · 𝑧))))
66	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑦)) = (𝐹‘(𝑢 · 𝑦)))
67	36, 39, 41, 66	syl3anc 1374	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑦)) = (𝐹‘(𝑢 · 𝑦)))
68	50	oveq2d 7376	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑦)) = (𝑢( ·𝑠 ‘𝑁)𝑣))
69	67, 68	eqtr3d 2774	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘(𝑢 · 𝑦)) = (𝑢( ·𝑠 ‘𝑁)𝑣))
70	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑢 · 𝑧)))
71	36, 39, 42, 70	syl3anc 1374	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑢 · 𝑧)))
72	51	oveq2d 7376	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝑢( ·𝑠 ‘𝑁)𝑤))
73	71, 72	eqtr3d 2774	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘(𝑢 · 𝑧)) = (𝑢( ·𝑠 ‘𝑁)𝑤))
74	69, 73	oveq12d 7378	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → ((𝐹‘(𝑢 · 𝑦))(+g‘𝑁)(𝐹‘(𝑢 · 𝑧))) = ((𝑢( ·𝑠 ‘𝑁)𝑣)(+g‘𝑁)(𝑢( ·𝑠 ‘𝑁)𝑤)))
75	59, 65, 74	3eqtr2d 2778	. . . 4 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝑢( ·𝑠 ‘𝑁)(𝑣(+g‘𝑁)𝑤)) = ((𝑢( ·𝑠 ‘𝑁)𝑣)(+g‘𝑁)(𝑢( ·𝑠 ‘𝑁)𝑤)))
76		simplll 775	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
77	37	simp2d 1144	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑣 ∈ 𝐵)
78		fofn 6748	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
79	4, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑉)
80		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
81		forn 6749	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
82	4, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
83	82	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → ran 𝐹 = 𝐵)
84	80, 83	eleqtrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran 𝐹)
85		fvelrnb 6894	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
86	85	biimpa 476	. . . . . 6 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑣 ∈ ran 𝐹) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
87	79, 84, 86	syl2an2r 686	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
88	76, 77, 87	syl2anc 585	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
89	75, 88	r19.29a 3146	. . 3 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝑣(+g‘𝑁)𝑤)) = ((𝑢( ·𝑠 ‘𝑁)𝑣)(+g‘𝑁)(𝑢( ·𝑠 ‘𝑁)𝑤)))
90		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
91	82	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ran 𝐹 = 𝐵)
92	90, 91	eleqtrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ ran 𝐹)
93		fvelrnb 6894	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
94	93	biimpa 476	. . . . 5 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑤 ∈ ran 𝐹) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
95	79, 92, 94	syl2an2r 686	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
96	95	3ad2antr3 1192	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
97	89, 96	r19.29a 3146	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢( ·𝑠 ‘𝑁)(𝑣(+g‘𝑁)𝑤)) = ((𝑢( ·𝑠 ‘𝑁)𝑣)(+g‘𝑁)(𝑢( ·𝑠 ‘𝑁)𝑤)))
98		simplll 775	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
99	5	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑀 ∈ LMod)
100		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵))
101	100	simp1d 1143	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑢 ∈ 𝑆)
102	100	simp2d 1144	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑣 ∈ 𝑆)
103		eqid 2737	. . . . . . 7 ⊢ (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
104	8, 11, 103	lmodacl 20858	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢(+g‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
105	99, 101, 102, 104	syl3anc 1374	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(+g‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
106		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑧 ∈ 𝑉)
107	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . 5 ⊢ ((𝜑 ∧ (𝑢(+g‘(Scalar‘𝑀))𝑣) ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧)))
108	98, 105, 106, 107	syl3anc 1374	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧)))
109		simpr 484	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
110	109	oveq2d 7376	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤))
111	2, 18, 8, 26, 11, 103	lmodvsdir 20872	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉)) → ((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧) = ((𝑢 · 𝑧) + (𝑣 · 𝑧)))
112	99, 101, 102, 106, 111	syl13anc 1375	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧) = ((𝑢 · 𝑧) + (𝑣 · 𝑧)))
113	112	fveq2d 6838	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧)) = (𝐹‘((𝑢 · 𝑧) + (𝑣 · 𝑧))))
114	99, 101, 106, 62	syl3anc 1374	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 · 𝑧) ∈ 𝑉)
115	2, 8, 26, 11	lmodvscl 20864	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑣 · 𝑧) ∈ 𝑉)
116	99, 102, 106, 115	syl3anc 1374	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑣 · 𝑧) ∈ 𝑉)
117	4, 20, 1, 3, 5, 18, 47	imasaddval 17487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 · 𝑧) ∈ 𝑉 ∧ (𝑣 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑢 · 𝑧))(+g‘𝑁)(𝐹‘(𝑣 · 𝑧))) = (𝐹‘((𝑢 · 𝑧) + (𝑣 · 𝑧))))
118	98, 114, 116, 117	syl3anc 1374	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘(𝑢 · 𝑧))(+g‘𝑁)(𝐹‘(𝑣 · 𝑧))) = (𝐹‘((𝑢 · 𝑧) + (𝑣 · 𝑧))))
119	98, 101, 106, 70	syl3anc 1374	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑢 · 𝑧)))
120	109	oveq2d 7376	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝑢( ·𝑠 ‘𝑁)𝑤))
121	119, 120	eqtr3d 2774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘(𝑢 · 𝑧)) = (𝑢( ·𝑠 ‘𝑁)𝑤))
122	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑣 · 𝑧)))
123	98, 102, 106, 122	syl3anc 1374	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑣 · 𝑧)))
124	109	oveq2d 7376	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝑣( ·𝑠 ‘𝑁)𝑤))
125	123, 124	eqtr3d 2774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘(𝑣 · 𝑧)) = (𝑣( ·𝑠 ‘𝑁)𝑤))
126	121, 125	oveq12d 7378	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘(𝑢 · 𝑧))(+g‘𝑁)(𝐹‘(𝑣 · 𝑧))) = ((𝑢( ·𝑠 ‘𝑁)𝑤)(+g‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
127	113, 118, 126	3eqtr2d 2778	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘((𝑢(+g‘(Scalar‘𝑀))𝑣) · 𝑧)) = ((𝑢( ·𝑠 ‘𝑁)𝑤)(+g‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
128	108, 110, 127	3eqtr3d 2780	. . 3 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤) = ((𝑢( ·𝑠 ‘𝑁)𝑤)(+g‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
129	95	3ad2antr3 1192	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
130	128, 129	r19.29a 3146	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤) = ((𝑢( ·𝑠 ‘𝑁)𝑤)(+g‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
131		eqid 2737	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
132	8, 11, 131	lmodmcl 20859	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢(.r‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
133	99, 101, 102, 132	syl3anc 1374	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
134	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢(.r‘(Scalar‘𝑀))𝑣) ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧)))
135	98, 133, 106, 134	syl3anc 1374	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧)))
136	109	oveq2d 7376	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤))
137	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ (𝑣 · 𝑧) ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑣 · 𝑧))) = (𝐹‘(𝑢 · (𝑣 · 𝑧))))
138	98, 101, 116, 137	syl3anc 1374	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑣 · 𝑧))) = (𝐹‘(𝑢 · (𝑣 · 𝑧))))
139	123	oveq2d 7376	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧))) = (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑣 · 𝑧))))
140	2, 8, 26, 11, 131	lmodvsass 20873	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉)) → ((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧) = (𝑢 · (𝑣 · 𝑧)))
141	99, 101, 102, 106, 140	syl13anc 1375	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧) = (𝑢 · (𝑣 · 𝑧)))
142	141	fveq2d 6838	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧)) = (𝐹‘(𝑢 · (𝑣 · 𝑧))))
143	138, 139, 142	3eqtr4rd 2783	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘((𝑢(.r‘(Scalar‘𝑀))𝑣) · 𝑧)) = (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧))))
144	135, 136, 143	3eqtr3d 2780	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤) = (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧))))
145	124	oveq2d 7376	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧))) = (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
146	144, 145	eqtrd 2772	. . 3 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤) = (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
147	146, 129	r19.29a 3146	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(.r‘(Scalar‘𝑀))𝑣)( ·𝑠 ‘𝑁)𝑤) = (𝑢( ·𝑠 ‘𝑁)(𝑣( ·𝑠 ‘𝑁)𝑤)))
148		simplll 775	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝜑)
149		eqid 2737	. . . . . . . 8 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
150	11, 149	ringidcl 20237	. . . . . . 7 ⊢ ((Scalar‘𝑀) ∈ Ring → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
151	17, 150	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
152	151	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
153		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝑥 ∈ 𝑉)
154	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17493	. . . . 5 ⊢ ((𝜑 ∧ (1r‘(Scalar‘𝑀)) ∈ 𝑆 ∧ 𝑥 ∈ 𝑉) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)(𝐹‘𝑥)) = (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)))
155	148, 152, 153, 154	syl3anc 1374	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)(𝐹‘𝑥)) = (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)))
156		simpr 484	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘𝑥) = 𝑢)
157	156	oveq2d 7376	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)(𝐹‘𝑥)) = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)𝑢))
158	5	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝑀 ∈ LMod)
159	2, 8, 26, 149	lmodvs1 20876	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ 𝑉) → ((1r‘(Scalar‘𝑀)) · 𝑥) = 𝑥)
160	158, 153, 159	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀)) · 𝑥) = 𝑥)
161	160	fveq2d 6838	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)) = (𝐹‘𝑥))
162	161, 156	eqtrd 2772	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)) = 𝑢)
163	155, 157, 162	3eqtr3d 2780	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)𝑢) = 𝑢)
164		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
165	82	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ran 𝐹 = 𝐵)
166	164, 165	eleqtrrd 2840	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ ran 𝐹)
167		fvelrnb 6894	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
168	167	biimpa 476	. . . 4 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑢 ∈ ran 𝐹) → ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)
169	79, 166, 168	syl2an2r 686	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)
170	163, 169	r19.29a 3146	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)𝑢) = 𝑢)
171	6, 7, 9, 10, 12, 13, 14, 15, 17, 25, 35, 97, 130, 147, 170	islmodd 20852	1 ⊢ (𝜑 → 𝑁 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ran crn 5625   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  0_gc0g 17393   “_s cimas 17459  Grpcgrp 18900  1_rcur 20153  Ringcrg 20205  LModclmod 20846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-0g 17395  df-imas 17463  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-mgp 20113  df-ur 20154  df-ring 20207  df-lmod 20848

This theorem is referenced by:  imaslmhm  33432  quslmod  33433