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Theorem srgmulgass 20157
Description: An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgmulgass.b 𝐵 = (Base‘𝑅)
srgmulgass.m · = (.g𝑅)
srgmulgass.t × = (.r𝑅)
Assertion
Ref Expression
srgmulgass ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Proof of Theorem srgmulgass
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7427 . . . . . . . 8 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 7435 . . . . . . 7 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 7427 . . . . . . 7 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2744 . . . . . 6 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
54imbi2d 340 . . . . 5 (𝑥 = 0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))))
6 oveq1 7427 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
76oveq1d 7435 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8 oveq1 7427 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
97, 8eqeq12d 2744 . . . . . 6 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
109imbi2d 340 . . . . 5 (𝑥 = 𝑦 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)))))
11 oveq1 7427 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
1211oveq1d 7435 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13 oveq1 7427 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1412, 13eqeq12d 2744 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
1514imbi2d 340 . . . . 5 (𝑥 = (𝑦 + 1) → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
16 oveq1 7427 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1716oveq1d 7435 . . . . . . 7 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18 oveq1 7427 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
1917, 18eqeq12d 2744 . . . . . 6 (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
2019imbi2d 340 . . . . 5 (𝑥 = 𝑁 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
21 simpr 484 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22 simpr 484 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
2322adantr 480 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑌𝐵)
24 srgmulgass.b . . . . . . . 8 𝐵 = (Base‘𝑅)
25 srgmulgass.t . . . . . . . 8 × = (.r𝑅)
26 eqid 2728 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2724, 25, 26srglz 20148 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
2821, 23, 27syl2anc 583 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0g𝑅) × 𝑌) = (0g𝑅))
29 simpl 482 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3029adantr 480 . . . . . . . 8 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑋𝐵)
31 srgmulgass.m . . . . . . . . 9 · = (.g𝑅)
3224, 26, 31mulg0 19030 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
3330, 32syl 17 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g𝑅))
3433oveq1d 7435 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3524, 25srgcl 20133 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3621, 30, 23, 35syl3anc 1369 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
3724, 26, 31mulg0 19030 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3836, 37syl 17 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3928, 34, 383eqtr4d 2778 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40 srgmnd 20130 . . . . . . . . . . . . . 14 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
4140adantl 481 . . . . . . . . . . . . 13 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
4241adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43 simpl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
4430adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋𝐵)
45 eqid 2728 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
4624, 31, 45mulgnn0p1 19040 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4742, 43, 44, 46syl3anc 1369 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4847oveq1d 7435 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌))
4921adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
5024, 31, 42, 43, 44mulgnn0cld 19050 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
5123adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌𝐵)
5224, 45, 25srgdir 20138 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5349, 50, 44, 51, 52syl13anc 1370 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5448, 53eqtrd 2768 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5554adantr 480 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
56 oveq1 7427 . . . . . . . . 9 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
57353expb 1118 . . . . . . . . . . . . 13 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
5857ancoms 458 . . . . . . . . . . . 12 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
5958adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
6024, 31, 45mulgnn0p1 19040 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6142, 43, 59, 60syl3anc 1369 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6261eqcomd 2734 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6356, 62sylan9eqr 2790 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6455, 63eqtrd 2768 . . . . . . 7 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6564exp31 419 . . . . . 6 (𝑦 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
6665a2d 29 . . . . 5 (𝑦 ∈ ℕ0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
675, 10, 15, 20, 39, 66nn0ind 12688 . . . 4 (𝑁 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
6867expd 415 . . 3 (𝑁 ∈ ℕ0 → ((𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
69683impib 1114 . 2 ((𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
7069impcom 407 1 ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  0cc0 11139  1c1 11140   + caddc 11142  0cn0 12503  Basecbs 17180  +gcplusg 17233  .rcmulr 17234  0gc0g 17421  Mndcmnd 18694  .gcmg 19023  SRingcsrg 20126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-seq 14000  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-plusg 17246  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mulg 19024  df-cmn 19737  df-mgp 20075  df-srg 20127
This theorem is referenced by:  srgpcomppsc  20160  srgbinomlem4  20169
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