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Theorem mulgass2 20323

Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)

**Hypotheses**
Ref	Expression
mulgass2.b	⊢ 𝐵 = (Base‘𝑅)
mulgass2.m	⊢ · = (._g‘𝑅)
mulgass2.t	⊢ × = (._r‘𝑅)

**Assertion**
Ref	Expression
mulgass2	⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Proof of Theorem mulgass2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
2	1	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
4	2, 3	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
6	5	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
7		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
8	6, 7	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
9		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
10	9	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
11		oveq1 7438	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
12	10, 11	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
13		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = -𝑦 → (𝑥 · 𝑋) = (-𝑦 · 𝑋))
14	13	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = -𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((-𝑦 · 𝑋) × 𝑌))
15		oveq1 7438	. . . . . 6 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑋 × 𝑌)) = (-𝑦 · (𝑋 × 𝑌)))
16	14, 15	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = -𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
17		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
18	17	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
19		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
20	18, 19	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
21		mulgass2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
22		mulgass2.t	. . . . . . . 8 ⊢ × = (._r‘𝑅)
23		eqid 2735	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
24	21, 22, 23	ringlz 20307	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ((0_g‘𝑅) × 𝑌) = (0_g‘𝑅))
25	24	3adant3 1131	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((0_g‘𝑅) × 𝑌) = (0_g‘𝑅))
26		simp3 1137	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
27		mulgass2.m	. . . . . . . . 9 ⊢ · = (._g‘𝑅)
28	21, 23, 27	mulg0 19105	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0_g‘𝑅))
29	26, 28	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0_g‘𝑅))
30	29	oveq1d 7446	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) × 𝑌) = ((0_g‘𝑅) × 𝑌))
31	21, 22	ringcl 20268	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
32	31	3com23 1125	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
33	21, 23, 27	mulg0 19105	. . . . . . 7 ⊢ ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0_g‘𝑅))
34	32, 33	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (0 · (𝑋 × 𝑌)) = (0_g‘𝑅))
35	25, 30, 34	3eqtr4d 2785	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
36		oveq1 7438	. . . . . . 7 ⊢ (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+_g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+_g‘𝑅)(𝑋 × 𝑌)))
37		simpl1 1190	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑅 ∈ Ring)
38		ringgrp 20256	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	37, 38	syl 17	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑅 ∈ Grp)
40		nn0z 12636	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
41	40	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
42	26	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑋 ∈ 𝐵)
43		eqid 2735	. . . . . . . . . . . 12 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	21, 27, 43	mulgp1 19138	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+_g‘𝑅)𝑋))
45	39, 41, 42, 44	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+_g‘𝑅)𝑋))
46	45	oveq1d 7446	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+_g‘𝑅)𝑋) × 𝑌))
47	38	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp)
48	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑅 ∈ Grp)
49	21, 27	mulgcl 19122	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
50	48, 41, 42, 49	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (𝑦 · 𝑋) ∈ 𝐵)
51		simpl2 1191	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → 𝑌 ∈ 𝐵)
52	21, 43, 22	ringdir 20279	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑦 · 𝑋)(+_g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+_g‘𝑅)(𝑋 × 𝑌)))
53	37, 50, 42, 51, 52	syl13anc 1371	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (((𝑦 · 𝑋)(+_g‘𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+_g‘𝑅)(𝑋 × 𝑌)))
54	46, 53	eqtrd 2775	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+_g‘𝑅)(𝑋 × 𝑌)))
55	32	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (𝑋 × 𝑌) ∈ 𝐵)
56	21, 27, 43	mulgp1 19138	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+_g‘𝑅)(𝑋 × 𝑌)))
57	39, 41, 55, 56	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+_g‘𝑅)(𝑋 × 𝑌)))
58	54, 57	eqeq12d 2751	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → ((((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)) ↔ (((𝑦 · 𝑋) × 𝑌)(+_g‘𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+_g‘𝑅)(𝑋 × 𝑌))))
59	36, 58	imbitrrid 246	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ₀) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
60	59	ex 412	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ₀ → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
61		fveq2 6907	. . . . . . 7 ⊢ (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((inv_g‘𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((inv_g‘𝑅)‘(𝑦 · (𝑋 × 𝑌))))
62	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Grp)
63		nnz 12632	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
64	63	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
65	26	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → 𝑋 ∈ 𝐵)
66		eqid 2735	. . . . . . . . . . . 12 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
67	21, 27, 66	mulgneg 19123	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑦 · 𝑋) = ((inv_g‘𝑅)‘(𝑦 · 𝑋)))
68	62, 64, 65, 67	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · 𝑋) = ((inv_g‘𝑅)‘(𝑦 · 𝑋)))
69	68	oveq1d 7446	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = (((inv_g‘𝑅)‘(𝑦 · 𝑋)) × 𝑌))
70		simpl1 1190	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Ring)
71	62, 64, 65, 49	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (𝑦 · 𝑋) ∈ 𝐵)
72		simpl2 1191	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → 𝑌 ∈ 𝐵)
73	21, 22, 66, 70, 71, 72	ringmneg1 20318	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (((inv_g‘𝑅)‘(𝑦 · 𝑋)) × 𝑌) = ((inv_g‘𝑅)‘((𝑦 · 𝑋) × 𝑌)))
74	69, 73	eqtrd 2775	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = ((inv_g‘𝑅)‘((𝑦 · 𝑋) × 𝑌)))
75	32	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (𝑋 × 𝑌) ∈ 𝐵)
76	21, 27, 66	mulgneg 19123	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → (-𝑦 · (𝑋 × 𝑌)) = ((inv_g‘𝑅)‘(𝑦 · (𝑋 × 𝑌))))
77	62, 64, 75, 76	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · (𝑋 × 𝑌)) = ((inv_g‘𝑅)‘(𝑦 · (𝑋 × 𝑌))))
78	74, 77	eqeq12d 2751	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌)) ↔ ((inv_g‘𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((inv_g‘𝑅)‘(𝑦 · (𝑋 × 𝑌)))))
79	61, 78	imbitrrid 246	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ ℕ) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
80	79	ex 412	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌)))))
81	4, 8, 12, 16, 20, 35, 60, 80	zindd 12717	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
82	81	3exp 1118	. . 3 ⊢ (𝑅 ∈ Ring → (𝑌 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))))
83	82	com24 95	. 2 ⊢ (𝑅 ∈ Ring → (𝑁 ∈ ℤ → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))))
84	83	3imp2 1348	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 -cneg 11491 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 Basecbs 17245 +_gcplusg 17298 ._rcmulr 17299 0_gc0g 17486 Grpcgrp 18964 inv_gcminusg 18965 ._gcmg 19098 Ringcrg 20251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-mulg 19099 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253

This theorem is referenced by: mulgass3 20370 mulgrhm 21506 dvdschrmulg 21561 zlmassa 21941 psdmul 22188 elrgspnlem2 33233 isarchiofld 33327 elrspunidl 33436

W3C validator