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Theorem mulgass2 19282
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 · = (.g𝑅)
mulgass2.t × = (.r𝑅)
Assertion
Ref Expression
mulgass2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Proof of Theorem mulgass2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7152 . . . . . . 7 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 7160 . . . . . 6 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 7152 . . . . . 6 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2837 . . . . 5 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5 oveq1 7152 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
65oveq1d 7160 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
7 oveq1 7152 . . . . . 6 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
86, 7eqeq12d 2837 . . . . 5 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
9 oveq1 7152 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
109oveq1d 7160 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
11 oveq1 7152 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1210, 11eqeq12d 2837 . . . . 5 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
13 oveq1 7152 . . . . . . 7 (𝑥 = -𝑦 → (𝑥 · 𝑋) = (-𝑦 · 𝑋))
1413oveq1d 7160 . . . . . 6 (𝑥 = -𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((-𝑦 · 𝑋) × 𝑌))
15 oveq1 7152 . . . . . 6 (𝑥 = -𝑦 → (𝑥 · (𝑋 × 𝑌)) = (-𝑦 · (𝑋 × 𝑌)))
1614, 15eqeq12d 2837 . . . . 5 (𝑥 = -𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
17 oveq1 7152 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1817oveq1d 7160 . . . . . 6 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
19 oveq1 7152 . . . . . 6 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
2018, 19eqeq12d 2837 . . . . 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 × = (.r𝑅)
23 eqid 2821 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2421, 22, 23ringlz 19268 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
25243adant3 1124 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
26 simp3 1130 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → 𝑋𝐵)
27 mulgass2.m . . . . . . . . 9 · = (.g𝑅)
2821, 23, 27mulg0 18171 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
2926, 28syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (0 · 𝑋) = (0g𝑅))
3029oveq1d 7160 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3121, 22ringcl 19242 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
32313com23 1118 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3321, 23, 27mulg0 18171 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3432, 33syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3525, 30, 343eqtr4d 2866 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
36 oveq1 7152 . . . . . . 7 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
37 simpl1 1183 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Ring)
38 ringgrp 19233 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Grp)
40 nn0z 11994 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℤ)
4140adantl 482 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℤ)
4226adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑋𝐵)
43 eqid 2821 . . . . . . . . . . . 12 (+g𝑅) = (+g𝑅)
4421, 27, 43mulgp1 18200 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4539, 41, 42, 44syl3anc 1363 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4645oveq1d 7160 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌))
47383ad2ant1 1125 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → 𝑅 ∈ Grp)
4847adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Grp)
4921, 27mulgcl 18185 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
5048, 41, 42, 49syl3anc 1363 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (𝑦 · 𝑋) ∈ 𝐵)
51 simpl2 1184 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑌𝐵)
5221, 43, 22ringdir 19248 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5337, 50, 42, 51, 52syl13anc 1364 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5446, 53eqtrd 2856 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5532adantr 481 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (𝑋 × 𝑌) ∈ 𝐵)
5621, 27, 43mulgp1 18200 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
5739, 41, 55, 56syl3anc 1363 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
5854, 57eqeq12d 2837 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → ((((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)) ↔ (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌))))
5936, 58syl5ibr 247 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
6059ex 413 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑦 ∈ ℕ0 → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
61 fveq2 6664 . . . . . . 7 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
6247adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Grp)
63 nnz 11993 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
6463adantl 482 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
6526adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑋𝐵)
66 eqid 2821 . . . . . . . . . . . 12 (invg𝑅) = (invg𝑅)
6721, 27, 66mulgneg 18186 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → (-𝑦 · 𝑋) = ((invg𝑅)‘(𝑦 · 𝑋)))
6862, 64, 65, 67syl3anc 1363 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · 𝑋) = ((invg𝑅)‘(𝑦 · 𝑋)))
6968oveq1d 7160 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = (((invg𝑅)‘(𝑦 · 𝑋)) × 𝑌))
70 simpl1 1183 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Ring)
7162, 64, 65, 49syl3anc 1363 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (𝑦 · 𝑋) ∈ 𝐵)
72 simpl2 1184 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑌𝐵)
7321, 22, 66, 70, 71, 72ringmneg1 19277 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((invg𝑅)‘(𝑦 · 𝑋)) × 𝑌) = ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)))
7469, 73eqtrd 2856 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)))
7532adantr 481 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (𝑋 × 𝑌) ∈ 𝐵)
7621, 27, 66mulgneg 18186 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → (-𝑦 · (𝑋 × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
7762, 64, 75, 76syl3anc 1363 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · (𝑋 × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
7874, 77eqeq12d 2837 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌)) ↔ ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌)))))
7961, 78syl5ibr 247 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
8079ex 413 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑦 ∈ ℕ → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌)))))
814, 8, 12, 16, 20, 35, 60, 80zindd 12072 . . . 4 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
82813exp 1111 . . 3 (𝑅 ∈ Ring → (𝑌𝐵 → (𝑋𝐵 → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))))
8382com24 95 . 2 (𝑅 ∈ Ring → (𝑁 ∈ ℤ → (𝑋𝐵 → (𝑌𝐵 → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))))
84833imp2 1341 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6349  (class class class)co 7145  0cc0 10526  1c1 10527   + caddc 10529  -cneg 10860  cn 11627  0cn0 11886  cz 11970  Basecbs 16473  +gcplusg 16555  .rcmulr 16556  0gc0g 16703  Grpcgrp 18043  invgcminusg 18044  .gcmg 18164  Ringcrg 19228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-seq 13360  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-minusg 18047  df-mulg 18165  df-mgp 19171  df-ur 19183  df-ring 19230
This theorem is referenced by:  mulgass3  19318  mulgrhm  20575  zlmassa  20601  dvdschrmulg  30786  isarchiofld  30818
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