Theorem islmod 20700

Description: The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
islmod.v	⊢ 𝑉 = (Base‘𝑊)
islmod.a	⊢ + = (+g‘𝑊)
islmod.s	⊢ · = ( ·𝑠 ‘𝑊)
islmod.f	⊢ 𝐹 = (Scalar‘𝑊)
islmod.k	⊢ 𝐾 = (Base‘𝐹)
islmod.p	⊢ ⨣ = (+g‘𝐹)
islmod.t	⊢ × = (.r‘𝐹)
islmod.u	⊢ 1 = (1r‘𝐹)

Assertion

Ref	Expression
islmod	⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))

Distinct variable groups:   𝑟,𝑞,𝑤,𝑥,𝐹   𝐾,𝑞,𝑟,𝑤,𝑥   ⨣ ,𝑞,𝑟,𝑤,𝑥   𝑉,𝑞,𝑟,𝑤,𝑥   + ,𝑞,𝑟,𝑤,𝑥   1 ,𝑞,𝑟,𝑤,𝑥   × ,𝑞,𝑟,𝑤,𝑥   · ,𝑞,𝑟,𝑤,𝑥

Allowed substitution hints:   𝑊(𝑥,𝑤,𝑟,𝑞)

Proof of Theorem islmod

Dummy variables 𝑓 𝑎 𝑔 𝑘 𝑝 𝑠 𝑣 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
2		islmod.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
4		fveq2 6881	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (+g‘𝑔) = (+g‘𝑊))
5		islmod.a	. . . . . . 7 ⊢ + = (+g‘𝑊)
6	4, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝑔 = 𝑊 → (+g‘𝑔) = + )
7		fveq2 6881	. . . . . . . 8 ⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = (Scalar‘𝑊))
8		islmod.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = 𝐹)
10		fveq2 6881	. . . . . . . . 9 ⊢ (𝑔 = 𝑊 → ( ·𝑠 ‘𝑔) = ( ·𝑠 ‘𝑊))
11		islmod.s	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊)
12	10, 11	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑔 = 𝑊 → ( ·𝑠 ‘𝑔) = · )
13	12	sbceq1d 3774	. . . . . . 7 ⊢ (𝑔 = 𝑊 → ([( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
14	9, 13	sbceqbid 3776	. . . . . 6 ⊢ (𝑔 = 𝑊 → ([(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [𝐹 / 𝑓][ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
15	6, 14	sbceqbid 3776	. . . . 5 ⊢ (𝑔 = 𝑊 → ([(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ + / 𝑎][𝐹 / 𝑓][ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
16	3, 15	sbceqbid 3776	. . . 4 ⊢ (𝑔 = 𝑊 → ([(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
17	2	fvexi 6895	. . . . . 6 ⊢ 𝑉 ∈ V
18	5	fvexi 6895	. . . . . 6 ⊢ + ∈ V
19	8	fvexi 6895	. . . . . 6 ⊢ 𝐹 ∈ V
20		simp3 1135	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
21	20	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22		islmod.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹)
23	21, 22	eqtr4di 2782	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
24	20	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = (+g‘𝐹))
25		islmod.p	. . . . . . . . . 10 ⊢ ⨣ = (+g‘𝐹)
26	24, 25	eqtr4di 2782	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = ⨣ )
27	20	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = (.r‘𝐹))
28		islmod.t	. . . . . . . . . . . 12 ⊢ × = (.r‘𝐹)
29	27, 28	eqtr4di 2782	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = × )
30	29	sbceq1d 3774	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ × / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
31	28	fvexi 6895	. . . . . . . . . . . 12 ⊢ × ∈ V
32		oveq 7407	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = × → (𝑞𝑡𝑟) = (𝑞 × 𝑟))
33	32	oveq1d 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = × → ((𝑞𝑡𝑟)𝑠𝑤) = ((𝑞 × 𝑟)𝑠𝑤))
34	33	eqeq1d 2726	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = × → (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤))))
35	34	anbi1d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = × → ((((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))
36	35	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = × → ((((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))))
37	36	2ralbidv 3210	. . . . . . . . . . . . . 14 ⊢ (𝑡 = × → (∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))))
38	37	2ralbidv 3210	. . . . . . . . . . . . 13 ⊢ (𝑡 = × → (∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))))
39	38	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑡 = × → ((𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))))
40	31, 39	sbcie 3812	. . . . . . . . . . 11 ⊢ ([ × / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))))
41	20	eleq1d 2810	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑓 ∈ Ring ↔ 𝐹 ∈ Ring))
42		simp1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑣 = 𝑉)
43	42	eleq2d 2811	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤) ∈ 𝑣 ↔ (𝑟𝑠𝑤) ∈ 𝑉))
44		simp2 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑎 = + )
45	44	oveqd 7418	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑤𝑎𝑥) = (𝑤 + 𝑥))
46	45	oveq2d 7417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑟𝑠(𝑤𝑎𝑥)) = (𝑟𝑠(𝑤 + 𝑥)))
47	44	oveqd 7418	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)))
48	46, 47	eqeq12d 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ↔ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥))))
49	44	oveqd 7418	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)))
50	49	eqeq2d 2735	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) ↔ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))))
51	43, 48, 50	3anbi123d 1432	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ↔ ((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)))))
52	20	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = (1r‘𝐹))
53		islmod.u	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (1r‘𝐹)
54	52, 53	eqtr4di 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = 1 )
55	54	oveq1d 7416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((1r‘𝑓)𝑠𝑤) = ( 1 𝑠𝑤))
56	55	eqeq1d 2726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((1r‘𝑓)𝑠𝑤) = 𝑤 ↔ ( 1 𝑠𝑤) = 𝑤))
57	56	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))
58	51, 57	anbi12d 630	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))
59	42, 58	raleqbidv 3334	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))
60	42, 59	raleqbidv 3334	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))
61	60	2ralbidv 3210	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))
62	41, 61	anbi12d 630	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
63	40, 62	bitrid 283	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([ × / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
64	30, 63	bitrd 279	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
65	26, 64	sbceqbid 3776	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
66	23, 65	sbceqbid 3776	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
67	66	sbcbidv 3828	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ · / 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))))
68	17, 18, 19, 67	sbc3ie 3855	. . . . 5 ⊢ ([𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ [ · / 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))
69	11	fvexi 6895	. . . . . 6 ⊢ · ∈ V
70	22	fvexi 6895	. . . . . 6 ⊢ 𝐾 ∈ V
71	25	fvexi 6895	. . . . . 6 ⊢ ⨣ ∈ V
72		simp2 1134	. . . . . . . 8 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑘 = 𝐾)
73		simp1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑠 = · )
74	73	oveqd 7418	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑤) = (𝑟 · 𝑤))
75	74	eleq1d 2810	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) ∈ 𝑉 ↔ (𝑟 · 𝑤) ∈ 𝑉))
76	73	oveqd 7418	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠(𝑤 + 𝑥)) = (𝑟 · (𝑤 + 𝑥)))
77	73	oveqd 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
78	74, 77	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)))
79	76, 78	eqeq12d 2740	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ↔ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥))))
80		simp3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑝 = ⨣ )
81	80	oveqd 7418	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑝𝑟) = (𝑞 ⨣ 𝑟))
82	81	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟)𝑠𝑤))
83	73	oveqd 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 ⨣ 𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤))
84	82, 83	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤))
85	73	oveqd 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠𝑤) = (𝑞 · 𝑤))
86	85, 74	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) = ((𝑞 · 𝑤) + (𝑟 · 𝑤)))
87	84, 86	eqeq12d 2740	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) ↔ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))))
88	75, 79, 87	3anbi123d 1432	. . . . . . . . . . 11 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ↔ ((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤)))))
89	73	oveqd 7418	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 × 𝑟)𝑠𝑤) = ((𝑞 × 𝑟) · 𝑤))
90	74	oveq2d 7417	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞𝑠(𝑟 · 𝑤)))
91	73	oveqd 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟 · 𝑤)) = (𝑞 · (𝑟 · 𝑤)))
92	90, 91	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞 · (𝑟 · 𝑤)))
93	89, 92	eqeq12d 2740	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤))))
94	73	oveqd 7418	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ( 1 𝑠𝑤) = ( 1 · 𝑤))
95	94	eqeq1d 2726	. . . . . . . . . . . 12 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (( 1 𝑠𝑤) = 𝑤 ↔ ( 1 · 𝑤) = 𝑤))
96	93, 95	anbi12d 630	. . . . . . . . . . 11 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))
97	88, 96	anbi12d 630	. . . . . . . . . 10 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
98	97	2ralbidv 3210	. . . . . . . . 9 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
99	72, 98	raleqbidv 3334	. . . . . . . 8 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
100	72, 99	raleqbidv 3334	. . . . . . 7 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
101	100	anbi2d 628	. . . . . 6 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))))
102	69, 70, 71, 101	sbc3ie 3855	. . . . 5 ⊢ ([ · / 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
103	68, 102	bitri 275	. . . 4 ⊢ ([𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
104	16, 103	bitrdi 287	. . 3 ⊢ (𝑔 = 𝑊 → ([(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))))
105		df-lmod 20698	. . 3 ⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))}
106	104, 105	elrab2 3678	. 2 ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))))
107		3anass 1092	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))) ↔ (𝑊 ∈ Grp ∧ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))))
108	106, 107	bitr4i 278	1 ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  [wsbc 3769  ‘cfv 6533  (class class class)co 7401  Basecbs 17143  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  Grpcgrp 18853  1_rcur 20076  Ringcrg 20128  LModclmod 20696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-lmod 20698

This theorem is referenced by:  lmodlema  20701  islmodd  20702  lmodgrp  20703  lmodring  20704  lmodprop2d  20760  isclmp  24946  lmodslmd  32817  ccfldsrarelvec  33225  lmod1  47361