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Theorem srgmulgass 18354
 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 6556 . . . . . . . 8 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 6564 . . . . . . 7 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 6556 . . . . . . 7 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2625 . . . . . 6 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
54imbi2d 329 . . . . 5 (𝑥 = 0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))))
6 oveq1 6556 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
76oveq1d 6564 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8 oveq1 6556 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
97, 8eqeq12d 2625 . . . . . 6 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
109imbi2d 329 . . . . 5 (𝑥 = 𝑦 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)))))
11 oveq1 6556 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
1211oveq1d 6564 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13 oveq1 6556 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1412, 13eqeq12d 2625 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
1514imbi2d 329 . . . . 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 6556 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1716oveq1d 6564 . . . . . . 7 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18 oveq1 6556 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
1917, 18eqeq12d 2625 . . . . . 6 (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
2019imbi2d 329 . . . . 5 (𝑥 = 𝑁 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
21 simpr 476 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22 simpr 476 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
2322adantr 480 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑌𝐵)
24 srgmulgass.b . . . . . . . 8 𝐵 = (Base‘𝑅)
25 srgmulgass.t . . . . . . . 8 × = (.r𝑅)
26 eqid 2610 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2724, 25, 26srglz 18350 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
2821, 23, 27syl2anc 691 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0g𝑅) × 𝑌) = (0g𝑅))
29 simpl 472 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3029adantr 480 . . . . . . . 8 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑋𝐵)
31 srgmulgass.m . . . . . . . . 9 · = (.g𝑅)
3224, 26, 31mulg0 17369 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
3330, 32syl 17 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g𝑅))
3433oveq1d 6564 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3524, 25srgcl 18335 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3621, 30, 23, 35syl3anc 1318 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
3724, 26, 31mulg0 17369 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3836, 37syl 17 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3928, 34, 383eqtr4d 2654 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40 srgmnd 18332 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
4430adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋𝐵)
45 eqid 2610 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
4624, 31, 45mulgnn0p1 17375 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4742, 43, 44, 46syl3anc 1318 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4847oveq1d 6564 . . . . . . . . . 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, 31mulgnn0cl 17381 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
5142, 43, 44, 50syl3anc 1318 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
5223adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌𝐵)
5324, 45, 25srgdir 18340 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5449, 51, 44, 52, 53syl13anc 1320 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5548, 54eqtrd 2644 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5655adantr 480 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
57 oveq1 6556 . . . . . . . . 9 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
58353expb 1258 . . . . . . . . . . . . 13 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
5958ancoms 468 . . . . . . . . . . . 12 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
6059adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
6124, 31, 45mulgnn0p1 17375 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6242, 43, 60, 61syl3anc 1318 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6362eqcomd 2616 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6457, 63sylan9eqr 2666 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6556, 64eqtrd 2644 . . . . . . 7 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6665exp31 628 . . . . . 6 (𝑦 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
6766a2d 29 . . . . 5 (𝑦 ∈ ℕ0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
685, 10, 15, 20, 39, 67nn0ind 11348 . . . 4 (𝑁 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
6968expd 451 . . 3 (𝑁 ∈ ℕ0 → ((𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
70693impib 1254 . 2 ((𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
7170impcom 445 1 ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Mndcmnd 17117  .gcmg 17363  SRingcsrg 18328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mulg 17364  df-cmn 18018  df-mgp 18313  df-srg 18329 This theorem is referenced by:  srgpcomppsc  18357  srgbinomlem4  18366
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