Theorem imaslmod 32145

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 17394	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑁))
7		eqidd 2737	. 2 ⊢ (𝜑 → (+g‘𝑁) = (+g‘𝑁))
8		eqid 2736	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
9	1, 3, 4, 5, 8	imassca 17401	. 2 ⊢ (𝜑 → (Scalar‘𝑀) = (Scalar‘𝑁))
10		eqidd 2737	. 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 2737	. 2 ⊢ (𝜑 → (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀)))
14		eqidd 2737	. 2 ⊢ (𝜑 → (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀)))
15		eqidd 2737	. 2 ⊢ (𝜑 → (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
16	8	lmodring 20330	. . 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 20329	. . . . 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 18863	. . 3 ⊢ (𝜑 → (𝑁 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑁)))
25	24	simpld 495	. 2 ⊢ (𝜑 → 𝑁 ∈ Grp)
26		imaslmod.t	. . . 4 ⊢ · = ( ·𝑠 ‘𝑀)
27		eqid 2736	. . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑀 ∈ LMod)
30		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑘 ∈ 𝑆)
31		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
32	2, 8, 26, 11	lmodvscl 20339	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉) → (𝑘 · 𝑏) ∈ 𝑉)
33	29, 30, 31, 32	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉)) → (𝑘 · 𝑏) ∈ 𝑉)
34	1, 3, 4, 5, 8, 11, 26, 27, 28, 33	imasvscaf 17421	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑁):(𝑆 × 𝐵)⟶𝐵)
35	34	fovcld 31554	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵) → (𝑢( ·𝑠 ‘𝑁)𝑣) ∈ 𝐵)
36		simp-5l 783	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝜑)
37		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
38	37	simp1d 1142	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑢 ∈ 𝑆)
39	38	ad2antrr 724	. . . . . . 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 767	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑦 ∈ 𝑉)
42		simp-4r 782	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → 𝑧 ∈ 𝑉)
43	2, 18	grpcl 18756	. . . . . . . 8 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 + 𝑧) ∈ 𝑉)
44	40, 41, 42, 43	syl3anc 1371	. . . . . . 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 17420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ (𝑦 + 𝑧) ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑢 · (𝑦 + 𝑧))))
46	36, 39, 44, 45	syl3anc 1371	. . . . . 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 2736	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
48	4, 20, 1, 3, 5, 18, 47	imasaddval 17414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
49	36, 41, 42, 48	syl3anc 1371	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) ∧ 𝑦 ∈ 𝑉) ∧ (𝐹‘𝑦) = 𝑣) → (𝐹‘𝑦) = 𝑣)
51		simpllr 774	. . . . . . . . 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 7375	. . . . . . . 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 2778	. . . . . . 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 7373	. . . . . 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 20345	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑢 · (𝑦 + 𝑧)) = ((𝑢 · 𝑦) + (𝑢 · 𝑧)))
57	55, 39, 41, 42, 56	syl13anc 1372	. . . . . . 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 6846	. . . . . 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 2784	. . . . 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 20339	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉) → (𝑢 · 𝑦) ∈ 𝑉)
61	55, 39, 41, 60	syl3anc 1371	. . . . . 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 20339	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑢 · 𝑧) ∈ 𝑉)
63	55, 39, 42, 62	syl3anc 1371	. . . . . 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 17414	. . . . . 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 1371	. . . . 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 17420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑦)) = (𝐹‘(𝑢 · 𝑦)))
67	36, 39, 41, 66	syl3anc 1371	. . . . . . 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 7373	. . . . . . 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 2778	. . . . . 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 17420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑢 · 𝑧)))
71	36, 39, 42, 70	syl3anc 1371	. . . . . . 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 7373	. . . . . . 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 2778	. . . . . 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 7375	. . . . 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 2782	. . . 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 773	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
77	37	simp2d 1143	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑣 ∈ 𝐵)
78		fofn 6758	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
79	4, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑉)
80		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
81		forn 6759	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
82	4, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
83	82	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → ran 𝐹 = 𝐵)
84	80, 83	eleqtrrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran 𝐹)
85		fvelrnb 6903	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
86	85	biimpa 477	. . . . . 6 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑣 ∈ ran 𝐹) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
87	79, 84, 86	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐵) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
88	76, 77, 87	syl2anc 584	. . . 4 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)
89	75, 88	r19.29a 3159	. . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
91	82	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ran 𝐹 = 𝐵)
92	90, 91	eleqtrrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ ran 𝐹)
93		fvelrnb 6903	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
94	93	biimpa 477	. . . . 5 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑤 ∈ ran 𝐹) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
95	79, 92, 94	syl2an2r 683	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
96	95	3ad2antr3 1190	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
97	89, 96	r19.29a 3159	. 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 773	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
99	5	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑀 ∈ LMod)
100		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵))
101	100	simp1d 1142	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑢 ∈ 𝑆)
102	100	simp2d 1143	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → 𝑣 ∈ 𝑆)
103		eqid 2736	. . . . . . 7 ⊢ (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
104	8, 11, 103	lmodacl 20333	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢(+g‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
105	99, 101, 102, 104	syl3anc 1371	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(+g‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
106		simplr 767	. . . . 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 17420	. . . . 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 1371	. . . 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 485	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
110	109	oveq2d 7373	. . . 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 20346	. . . . . . 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 1372	. . . . . 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 6846	. . . . 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 1371	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢 · 𝑧) ∈ 𝑉)
115	2, 8, 26, 11	lmodvscl 20339	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑣 · 𝑧) ∈ 𝑉)
116	99, 102, 106, 115	syl3anc 1371	. . . . . 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 17414	. . . . . 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 1371	. . . . 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 1371	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑢 · 𝑧)))
120	109	oveq2d 7373	. . . . . . 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 2778	. . . . . 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 17420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉) → (𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑣 · 𝑧)))
123	98, 102, 106, 122	syl3anc 1371	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝑣( ·𝑠 ‘𝑁)(𝐹‘𝑧)) = (𝐹‘(𝑣 · 𝑧)))
124	109	oveq2d 7373	. . . . . . 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 2778	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) ∧ 𝑧 ∈ 𝑉) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘(𝑣 · 𝑧)) = (𝑣( ·𝑠 ‘𝑁)𝑤))
126	121, 125	oveq12d 7375	. . . . 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 2782	. . . 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 2784	. . 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 1190	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵)) → ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)
130	128, 129	r19.29a 3159	. 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 2736	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
132	8, 11, 131	lmodmcl 20334	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢(.r‘(Scalar‘𝑀))𝑣) ∈ 𝑆)
133	99, 101, 102, 132	syl3anc 1371	. . . . . 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 17420	. . . . . 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 1371	. . . . 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 7373	. . . . 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 17420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆 ∧ (𝑣 · 𝑧) ∈ 𝑉) → (𝑢( ·𝑠 ‘𝑁)(𝐹‘(𝑣 · 𝑧))) = (𝐹‘(𝑢 · (𝑣 · 𝑧))))
138	98, 101, 116, 137	syl3anc 1371	. . . . . 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 7373	. . . . . 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 20347	. . . . . . . 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 1372	. . . . . . 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 6846	. . . . . 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 2787	. . . . 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 2784	. . . 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 7373	. . . 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 2776	. . 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 3159	. 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 773	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝜑)
149		eqid 2736	. . . . . . . 8 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
150	11, 149	ringidcl 19989	. . . . . . 7 ⊢ ((Scalar‘𝑀) ∈ Ring → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
151	17, 150	syl 17	. . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
152	151	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (1r‘(Scalar‘𝑀)) ∈ 𝑆)
153		simplr 767	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝑥 ∈ 𝑉)
154	1, 3, 4, 5, 8, 11, 26, 27, 28	imasvscaval 17420	. . . . 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 1371	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)(𝐹‘𝑥)) = (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)))
156		simpr 485	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘𝑥) = 𝑢)
157	156	oveq2d 7373	. . . 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 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → 𝑀 ∈ LMod)
159	2, 8, 26, 149	lmodvs1 20350	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ 𝑉) → ((1r‘(Scalar‘𝑀)) · 𝑥) = 𝑥)
160	158, 153, 159	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀)) · 𝑥) = 𝑥)
161	160	fveq2d 6846	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)) = (𝐹‘𝑥))
162	161, 156	eqtrd 2776	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → (𝐹‘((1r‘(Scalar‘𝑀)) · 𝑥)) = 𝑢)
163	155, 157, 162	3eqtr3d 2784	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑥 ∈ 𝑉) ∧ (𝐹‘𝑥) = 𝑢) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑁)𝑢) = 𝑢)
164		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
165	82	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ran 𝐹 = 𝐵)
166	164, 165	eleqtrrd 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ ran 𝐹)
167		fvelrnb 6903	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
168	167	biimpa 477	. . . 4 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑢 ∈ ran 𝐹) → ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)
169	79, 166, 168	syl2an2r 683	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)
170	163, 169	r19.29a 3159	. 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 20328	1 ⊢ (𝜑 → 𝑁 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  ran crn 5634   Fn wfn 6491  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  ._rcmulr 17134  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321   “_s cimas 17386  Grpcgrp 18748  1_rcur 19913  Ringcrg 19964  LModclmod 20322

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-0g 17323  df-imas 17390  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324

This theorem is referenced by:  quslmod  32146