Theorem islmod 13847

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		elex 2774	. . . 4 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ V)
2		islmod.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
3		basfn 12736	. . . . . . . 8 ⊢ Base Fn V
4		funfvex 5575	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
5	4	funfni 5358	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
6	3, 5	mpan 424	. . . . . . 7 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
7	2, 6	eqeltrid 2283	. . . . . 6 ⊢ (𝑊 ∈ V → 𝑉 ∈ V)
8		islmod.a	. . . . . . . . 9 ⊢ + = (+g‘𝑊)
9		plusgslid 12790	. . . . . . . . . 10 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
10	9	slotex 12705	. . . . . . . . 9 ⊢ (𝑊 ∈ V → (+g‘𝑊) ∈ V)
11	8, 10	eqeltrid 2283	. . . . . . . 8 ⊢ (𝑊 ∈ V → + ∈ V)
12	11	adantr 276	. . . . . . 7 ⊢ ((𝑊 ∈ V ∧ 𝑣 = 𝑉) → + ∈ V)
13		islmod.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
14		scaslid 12830	. . . . . . . . . . . 12 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
15	14	slotex 12705	. . . . . . . . . . 11 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
16	13, 15	eqeltrid 2283	. . . . . . . . . 10 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
17	16	adantr 276	. . . . . . . . 9 ⊢ ((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) → 𝐹 ∈ V)
18		simplrl 535	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → 𝑣 = 𝑉)
19		simplrr 536	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → 𝑎 = + )
20		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
21		simp3 1001	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
22	21	fveq2d 5562	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
23		islmod.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘𝐹)
24	22, 23	eqtr4di 2247	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
25	18, 19, 20, 24	syl3anc 1249	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
26	21	fveq2d 5562	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = (+g‘𝐹))
27		islmod.p	. . . . . . . . . . . . . 14 ⊢ ⨣ = (+g‘𝐹)
28	26, 27	eqtr4di 2247	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = ⨣ )
29	18, 19, 20, 28	syl3anc 1249	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → (+g‘𝑓) = ⨣ )
30	21	fveq2d 5562	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = (.r‘𝐹))
31		islmod.t	. . . . . . . . . . . . . . . 16 ⊢ × = (.r‘𝐹)
32	30, 31	eqtr4di 2247	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = × )
33	32	sbceq1d 2994	. . . . . . . . . . . . . 14 ⊢ ((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)))))
34	18, 19, 20, 33	syl3anc 1249	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ V ∧ (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)))))
35		simpll 527	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ V ∧ (𝑣 = 𝑉 ∧ 𝑎 = + )) ∧ 𝑓 = 𝐹) → 𝑊 ∈ V)
36		mulrslid 12809	. . . . . . . . . . . . . . . . . . 19 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
37	36	slotex 12705	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ V → (.r‘𝐹) ∈ V)
38	16, 37	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ V → (.r‘𝐹) ∈ V)
39	31, 38	eqeltrid 2283	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ V → × ∈ V)
40		oveq 5928	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = × → (𝑞𝑡𝑟) = (𝑞 × 𝑟))
41	40	oveq1d 5937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = × → ((𝑞𝑡𝑟)𝑠𝑤) = ((𝑞 × 𝑟)𝑠𝑤))
42	41	eqeq1d 2205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = × → (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤))))
43	42	anbi1d 465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = × → ((((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))
44	43	anbi2d 464	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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))))
45	44	2ralbidv 2521	. . . . . . . . . . . . . . . . . . 19 ⊢ (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))))
46	45	2ralbidv 2521	. . . . . . . . . . . . . . . . . 18 ⊢ (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))))
47	46	anbi2d 464	. . . . . . . . . . . . . . . . 17 ⊢ (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)))))
48	47	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ V ∧ 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)))))
49	39, 48	sbcied 3026	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ V → ([ × / 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)))))
50	21	eleq1d 2265	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑓 ∈ Ring ↔ 𝐹 ∈ Ring))
51		simp1 999	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑣 = 𝑉)
52	51	eleq2d 2266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤) ∈ 𝑣 ↔ (𝑟𝑠𝑤) ∈ 𝑉))
53		simp2 1000	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑎 = + )
54	53	oveqd 5939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑤𝑎𝑥) = (𝑤 + 𝑥))
55	54	oveq2d 5938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑟𝑠(𝑤𝑎𝑥)) = (𝑟𝑠(𝑤 + 𝑥)))
56	53	oveqd 5939	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)))
57	55, 56	eqeq12d 2211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ↔ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥))))
58	53	oveqd 5939	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)))
59	58	eqeq2d 2208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) ↔ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))))
60	52, 57, 59	3anbi123d 1323	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)))))
61	21	fveq2d 5562	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = (1r‘𝐹))
62		islmod.u	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 = (1r‘𝐹)
63	61, 62	eqtr4di 2247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = 1 )
64	63	oveq1d 5937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((1r‘𝑓)𝑠𝑤) = ( 1 𝑠𝑤))
65	64	eqeq1d 2205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((1r‘𝑓)𝑠𝑤) = 𝑤 ↔ ( 1 𝑠𝑤) = 𝑤))
66	65	anbi2d 464	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))
67	60, 66	anbi12d 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))))
68	51, 67	raleqbidv 2709	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))))
69	51, 68	raleqbidv 2709	. . . . . . . . . . . . . . . . 17 ⊢ ((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))))
70	69	2ralbidv 2521	. . . . . . . . . . . . . . . 16 ⊢ ((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))))
71	50, 70	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ ((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)))))
72	49, 71	sylan9bb 462	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ V ∧ (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)))))
73	35, 18, 19, 20, 72	syl13anc 1251	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ V ∧ (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)))))
74	34, 73	bitrd 188	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ V ∧ (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)))))
75	29, 74	sbceqbid 2996	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ V ∧ (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)))))
76	25, 75	sbceqbid 2996	. . . . . . . . . 10 ⊢ (((𝑊 ∈ V ∧ (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)))))
77	76	sbcbidv 3048	. . . . . . . . 9 ⊢ (((𝑊 ∈ V ∧ (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)))))
78	17, 77	sbcied 3026	. . . . . . . 8 ⊢ ((𝑊 ∈ V ∧ (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)))))
79	78	anassrs 400	. . . . . . 7 ⊢ (((𝑊 ∈ V ∧ 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)))))
80	12, 79	sbcied 3026	. . . . . 6 ⊢ ((𝑊 ∈ V ∧ 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)))))
81	7, 80	sbcied 3026	. . . . 5 ⊢ (𝑊 ∈ V → ([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)))))
82		islmod.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
83		vscaslid 12840	. . . . . . . 8 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
84	83	slotex 12705	. . . . . . 7 ⊢ (𝑊 ∈ V → ( ·𝑠 ‘𝑊) ∈ V)
85	82, 84	eqeltrid 2283	. . . . . 6 ⊢ (𝑊 ∈ V → · ∈ V)
86		funfvex 5575	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
87	86	funfni 5358	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐹 ∈ V) → (Base‘𝐹) ∈ V)
88	3, 16, 87	sylancr 414	. . . . . . . . 9 ⊢ (𝑊 ∈ V → (Base‘𝐹) ∈ V)
89	23, 88	eqeltrid 2283	. . . . . . . 8 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
90	89	adantr 276	. . . . . . 7 ⊢ ((𝑊 ∈ V ∧ 𝑠 = · ) → 𝐾 ∈ V)
91	9	slotex 12705	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ V → (+g‘𝐹) ∈ V)
92	16, 91	syl 14	. . . . . . . . . . 11 ⊢ (𝑊 ∈ V → (+g‘𝐹) ∈ V)
93	27, 92	eqeltrid 2283	. . . . . . . . . 10 ⊢ (𝑊 ∈ V → ⨣ ∈ V)
94	93	adantr 276	. . . . . . . . 9 ⊢ ((𝑊 ∈ V ∧ (𝑠 = · ∧ 𝑘 = 𝐾)) → ⨣ ∈ V)
95		simp2 1000	. . . . . . . . . . . . 13 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑘 = 𝐾)
96		simp1 999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑠 = · )
97	96	oveqd 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑤) = (𝑟 · 𝑤))
98	97	eleq1d 2265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) ∈ 𝑉 ↔ (𝑟 · 𝑤) ∈ 𝑉))
99	96	oveqd 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠(𝑤 + 𝑥)) = (𝑟 · (𝑤 + 𝑥)))
100	96	oveqd 5939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
101	97, 100	oveq12d 5940	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)))
102	99, 101	eqeq12d 2211	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ↔ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥))))
103		simp3 1001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑝 = ⨣ )
104	103	oveqd 5939	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑝𝑟) = (𝑞 ⨣ 𝑟))
105	104	oveq1d 5937	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟)𝑠𝑤))
106	96	oveqd 5939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 ⨣ 𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤))
107	105, 106	eqtrd 2229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤))
108	96	oveqd 5939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠𝑤) = (𝑞 · 𝑤))
109	108, 97	oveq12d 5940	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) = ((𝑞 · 𝑤) + (𝑟 · 𝑤)))
110	107, 109	eqeq12d 2211	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) ↔ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))))
111	98, 102, 110	3anbi123d 1323	. . . . . . . . . . . . . . . 16 ⊢ ((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)))))
112	96	oveqd 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 × 𝑟)𝑠𝑤) = ((𝑞 × 𝑟) · 𝑤))
113	97	oveq2d 5938	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞𝑠(𝑟 · 𝑤)))
114	96	oveqd 5939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟 · 𝑤)) = (𝑞 · (𝑟 · 𝑤)))
115	113, 114	eqtrd 2229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞 · (𝑟 · 𝑤)))
116	112, 115	eqeq12d 2211	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤))))
117	96	oveqd 5939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ( 1 𝑠𝑤) = ( 1 · 𝑤))
118	117	eqeq1d 2205	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (( 1 𝑠𝑤) = 𝑤 ↔ ( 1 · 𝑤) = 𝑤))
119	116, 118	anbi12d 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))
120	111, 119	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ ((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))))
121	120	2ralbidv 2521	. . . . . . . . . . . . . 14 ⊢ ((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))))
122	95, 121	raleqbidv 2709	. . . . . . . . . . . . 13 ⊢ ((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))))
123	95, 122	raleqbidv 2709	. . . . . . . . . . . 12 ⊢ ((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))))
124	123	anbi2d 464	. . . . . . . . . . 11 ⊢ ((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)))))
125	124	3expa 1205	. . . . . . . . . 10 ⊢ (((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)))))
126	125	adantll 476	. . . . . . . . 9 ⊢ (((𝑊 ∈ V ∧ (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)))))
127	94, 126	sbcied 3026	. . . . . . . 8 ⊢ ((𝑊 ∈ V ∧ (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)))))
128	127	anassrs 400	. . . . . . 7 ⊢ (((𝑊 ∈ V ∧ 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)))))
129	90, 128	sbcied 3026	. . . . . 6 ⊢ ((𝑊 ∈ V ∧ 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)))))
130	85, 129	sbcied 3026	. . . . 5 ⊢ (𝑊 ∈ V → ([ · / 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)))))
131	81, 130	bitrd 188	. . . 4 ⊢ (𝑊 ∈ V → ([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)))))
132	1, 131	syl 14	. . 3 ⊢ (𝑊 ∈ Grp → ([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)))))
133	132	pm5.32i 454	. 2 ⊢ ((𝑊 ∈ Grp ∧ [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)))) ↔ (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)))))
134		fveq2 5558	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
135	134, 2	eqtr4di 2247	. . . 4 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
136		fveq2 5558	. . . . . 6 ⊢ (𝑔 = 𝑊 → (+g‘𝑔) = (+g‘𝑊))
137	136, 8	eqtr4di 2247	. . . . 5 ⊢ (𝑔 = 𝑊 → (+g‘𝑔) = + )
138		fveq2 5558	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = (Scalar‘𝑊))
139	138, 13	eqtr4di 2247	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = 𝐹)
140		fveq2 5558	. . . . . . . 8 ⊢ (𝑔 = 𝑊 → ( ·𝑠 ‘𝑔) = ( ·𝑠 ‘𝑊))
141	140, 82	eqtr4di 2247	. . . . . . 7 ⊢ (𝑔 = 𝑊 → ( ·𝑠 ‘𝑔) = · )
142	141	sbceq1d 2994	. . . . . 6 ⊢ (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)))))
143	139, 142	sbceqbid 2996	. . . . 5 ⊢ (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)))))
144	137, 143	sbceqbid 2996	. . . 4 ⊢ (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)))))
145	135, 144	sbceqbid 2996	. . 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))) ↔ [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)))))
146		df-lmod 13845	. . 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)))}
147	145, 146	elrab2 2923	. 2 ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ [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)))))
148		3anass 984	. 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)))))
149	133, 147, 148	3bitr4i 212	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:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  [wsbc 2989   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  ._rcmulr 12756  Scalarcsca 12758   ·_𝑠 cvsca 12759  Grpcgrp 13132  1_rcur 13515  Ringcrg 13552  LModclmod 13843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-lmod 13845

This theorem is referenced by:  lmodlema  13848  islmodd  13849  lmodgrp  13850  lmodring  13851  lmodprop2d  13904