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Theorem srgmulgass 19948
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 7364 . . . . . . . 8 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 7372 . . . . . . 7 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 7364 . . . . . . 7 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2752 . . . . . 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 7364 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
76oveq1d 7372 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8 oveq1 7364 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
97, 8eqeq12d 2752 . . . . . 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 7364 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
1211oveq1d 7372 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13 oveq1 7364 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1412, 13eqeq12d 2752 . . . . . 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 7364 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1716oveq1d 7372 . . . . . . 7 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18 oveq1 7364 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
1917, 18eqeq12d 2752 . . . . . 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 485 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22 simpr 485 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
2322adantr 481 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑌𝐵)
24 srgmulgass.b . . . . . . . 8 𝐵 = (Base‘𝑅)
25 srgmulgass.t . . . . . . . 8 × = (.r𝑅)
26 eqid 2736 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2724, 25, 26srglz 19939 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
2821, 23, 27syl2anc 584 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0g𝑅) × 𝑌) = (0g𝑅))
29 simpl 483 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3029adantr 481 . . . . . . . 8 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑋𝐵)
31 srgmulgass.m . . . . . . . . 9 · = (.g𝑅)
3224, 26, 31mulg0 18879 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
3330, 32syl 17 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g𝑅))
3433oveq1d 7372 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3524, 25srgcl 19924 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3621, 30, 23, 35syl3anc 1371 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
3724, 26, 31mulg0 18879 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3836, 37syl 17 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3928, 34, 383eqtr4d 2786 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40 srgmnd 19921 . . . . . . . . . . . . . 14 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
4140adantl 482 . . . . . . . . . . . . 13 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
4241adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43 simpl 483 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
4430adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋𝐵)
45 eqid 2736 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
4624, 31, 45mulgnn0p1 18887 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4742, 43, 44, 46syl3anc 1371 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4847oveq1d 7372 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌))
4921adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
5024, 31, 42, 43, 44mulgnn0cld 18897 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
5123adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌𝐵)
5224, 45, 25srgdir 19929 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5349, 50, 44, 51, 52syl13anc 1372 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5448, 53eqtrd 2776 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5554adantr 481 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
56 oveq1 7364 . . . . . . . . 9 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
57353expb 1120 . . . . . . . . . . . . 13 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
5857ancoms 459 . . . . . . . . . . . 12 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
5958adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
6024, 31, 45mulgnn0p1 18887 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6142, 43, 59, 60syl3anc 1371 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6261eqcomd 2742 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6356, 62sylan9eqr 2798 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6455, 63eqtrd 2776 . . . . . . 7 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6564exp31 420 . . . . . 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 12598 . . . 4 (𝑁 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
6867expd 416 . . 3 (𝑁 ∈ ℕ0 → ((𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
69683impib 1116 . 2 ((𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
7069impcom 408 1 ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  0cn0 12413  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  Mndcmnd 18556  .gcmg 18872  SRingcsrg 19917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mulg 18873  df-cmn 19564  df-mgp 19897  df-srg 19918
This theorem is referenced by:  srgpcomppsc  19951  srgbinomlem4  19960
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