MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgass2 Structured version   Visualization version   GIF version

Theorem mulgass2 18647
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 6697 . . . . . . 7 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 6705 . . . . . 6 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 6697 . . . . . 6 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2666 . . . . 5 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
5 oveq1 6697 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
65oveq1d 6705 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
7 oveq1 6697 . . . . . 6 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
86, 7eqeq12d 2666 . . . . 5 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
9 oveq1 6697 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
109oveq1d 6705 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
11 oveq1 6697 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1210, 11eqeq12d 2666 . . . . 5 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
13 oveq1 6697 . . . . . . 7 (𝑥 = -𝑦 → (𝑥 · 𝑋) = (-𝑦 · 𝑋))
1413oveq1d 6705 . . . . . 6 (𝑥 = -𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((-𝑦 · 𝑋) × 𝑌))
15 oveq1 6697 . . . . . 6 (𝑥 = -𝑦 → (𝑥 · (𝑋 × 𝑌)) = (-𝑦 · (𝑋 × 𝑌)))
1614, 15eqeq12d 2666 . . . . 5 (𝑥 = -𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
17 oveq1 6697 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1817oveq1d 6705 . . . . . 6 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
19 oveq1 6697 . . . . . 6 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
2018, 19eqeq12d 2666 . . . . 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 2651 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2421, 22, 23ringlz 18633 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
25243adant3 1101 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
26 simp3 1083 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → 𝑋𝐵)
27 mulgass2.m . . . . . . . . 9 · = (.g𝑅)
2821, 23, 27mulg0 17593 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
2926, 28syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (0 · 𝑋) = (0g𝑅))
3029oveq1d 6705 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3121, 22ringcl 18607 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
32313com23 1291 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3321, 23, 27mulg0 17593 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3432, 33syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3525, 30, 343eqtr4d 2695 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
36 oveq1 6697 . . . . . . 7 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
37 simpl1 1084 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Ring)
38 ringgrp 18598 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Grp)
40 nn0z 11438 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℤ)
4140adantl 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℤ)
4226adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑋𝐵)
43 eqid 2651 . . . . . . . . . . . 12 (+g𝑅) = (+g𝑅)
4421, 27, 43mulgp1 17621 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4539, 41, 42, 44syl3anc 1366 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4645oveq1d 6705 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌))
47383ad2ant1 1102 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → 𝑅 ∈ Grp)
4847adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ Grp)
4921, 27mulgcl 17606 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
5048, 41, 42, 49syl3anc 1366 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (𝑦 · 𝑋) ∈ 𝐵)
51 simpl2 1085 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → 𝑌𝐵)
5221, 43, 22ringdir 18613 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5337, 50, 42, 51, 52syl13anc 1368 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5446, 53eqtrd 2685 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5532adantr 480 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (𝑋 × 𝑌) ∈ 𝐵)
5621, 27, 43mulgp1 17621 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
5739, 41, 55, 56syl3anc 1366 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
5854, 57eqeq12d 2666 . . . . . . 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 236 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ0) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
6059ex 449 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑦 ∈ ℕ0 → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
61 fveq2 6229 . . . . . . 7 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
6247adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Grp)
63 nnz 11437 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
6463adantl 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
6526adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑋𝐵)
66 eqid 2651 . . . . . . . . . . . 12 (invg𝑅) = (invg𝑅)
6721, 27, 66mulgneg 17607 . . . . . . . . . . 11 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋𝐵) → (-𝑦 · 𝑋) = ((invg𝑅)‘(𝑦 · 𝑋)))
6862, 64, 65, 67syl3anc 1366 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · 𝑋) = ((invg𝑅)‘(𝑦 · 𝑋)))
6968oveq1d 6705 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = (((invg𝑅)‘(𝑦 · 𝑋)) × 𝑌))
70 simpl1 1084 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑅 ∈ Ring)
7162, 64, 65, 49syl3anc 1366 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (𝑦 · 𝑋) ∈ 𝐵)
72 simpl2 1085 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → 𝑌𝐵)
7321, 22, 66, 70, 71, 72ringmneg1 18642 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((invg𝑅)‘(𝑦 · 𝑋)) × 𝑌) = ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)))
7469, 73eqtrd 2685 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → ((-𝑦 · 𝑋) × 𝑌) = ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)))
7532adantr 480 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (𝑋 × 𝑌) ∈ 𝐵)
7621, 27, 66mulgneg 17607 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝑋 × 𝑌) ∈ 𝐵) → (-𝑦 · (𝑋 × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
7762, 64, 75, 76syl3anc 1366 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (-𝑦 · (𝑋 × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌))))
7874, 77eqeq12d 2666 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌)) ↔ ((invg𝑅)‘((𝑦 · 𝑋) × 𝑌)) = ((invg𝑅)‘(𝑦 · (𝑋 × 𝑌)))))
7961, 78syl5ibr 236 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) ∧ 𝑦 ∈ ℕ) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → ((-𝑦 · 𝑋) × 𝑌) = (-𝑦 · (𝑋 × 𝑌))))
8079ex 449 . . . . 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 11516 . . . 4 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
82813exp 1283 . . 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 1304 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  -cneg 10305  cn 11058  0cn0 11330  cz 11415  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470  .gcmg 17587  Ringcrg 18593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mulg 17588  df-mgp 18536  df-ur 18548  df-ring 18595
This theorem is referenced by:  mulgass3  18683  mulgrhm  19894  zlmassa  19920  isarchiofld  29945
  Copyright terms: Public domain W3C validator