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Theorem srgmulgass 18463
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 6617 . . . . . . . 8 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 6625 . . . . . . 7 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 6617 . . . . . . 7 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2636 . . . . . 6 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
54imbi2d 330 . . . . 5 (𝑥 = 0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))))
6 oveq1 6617 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
76oveq1d 6625 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8 oveq1 6617 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
97, 8eqeq12d 2636 . . . . . 6 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
109imbi2d 330 . . . . 5 (𝑥 = 𝑦 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)))))
11 oveq1 6617 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
1211oveq1d 6625 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13 oveq1 6617 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1412, 13eqeq12d 2636 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
1514imbi2d 330 . . . . 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 6617 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1716oveq1d 6625 . . . . . . 7 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18 oveq1 6617 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
1917, 18eqeq12d 2636 . . . . . 6 (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
2019imbi2d 330 . . . . 5 (𝑥 = 𝑁 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
21 simpr 477 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22 simpr 477 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
2322adantr 481 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑌𝐵)
24 srgmulgass.b . . . . . . . 8 𝐵 = (Base‘𝑅)
25 srgmulgass.t . . . . . . . 8 × = (.r𝑅)
26 eqid 2621 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2724, 25, 26srglz 18459 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
2821, 23, 27syl2anc 692 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0g𝑅) × 𝑌) = (0g𝑅))
29 simpl 473 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3029adantr 481 . . . . . . . 8 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑋𝐵)
31 srgmulgass.m . . . . . . . . 9 · = (.g𝑅)
3224, 26, 31mulg0 17478 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
3330, 32syl 17 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g𝑅))
3433oveq1d 6625 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3524, 25srgcl 18444 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3621, 30, 23, 35syl3anc 1323 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
3724, 26, 31mulg0 17478 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3836, 37syl 17 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3928, 34, 383eqtr4d 2665 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40 srgmnd 18441 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
4430adantl 482 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋𝐵)
45 eqid 2621 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
4624, 31, 45mulgnn0p1 17484 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4742, 43, 44, 46syl3anc 1323 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4847oveq1d 6625 . . . . . . . . . 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, 31mulgnn0cl 17490 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
5142, 43, 44, 50syl3anc 1323 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
5223adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌𝐵)
5324, 45, 25srgdir 18449 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5449, 51, 44, 52, 53syl13anc 1325 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5548, 54eqtrd 2655 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5655adantr 481 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
57 oveq1 6617 . . . . . . . . 9 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
58353expb 1263 . . . . . . . . . . . . 13 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
5958ancoms 469 . . . . . . . . . . . 12 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
6059adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
6124, 31, 45mulgnn0p1 17484 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6242, 43, 60, 61syl3anc 1323 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6362eqcomd 2627 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6457, 63sylan9eqr 2677 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6556, 64eqtrd 2655 . . . . . . 7 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6665exp31 629 . . . . . 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 11424 . . . 4 (𝑁 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
6968expd 452 . . 3 (𝑁 ∈ ℕ0 → ((𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
70693impib 1259 . 2 ((𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
7170impcom 446 1 ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5852  (class class class)co 6610  0cc0 9888  1c1 9889   + caddc 9891  0cn0 11244  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  0gc0g 16032  Mndcmnd 17226  .gcmg 17472  SRingcsrg 18437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-seq 12750  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-plusg 15886  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mulg 17473  df-cmn 18127  df-mgp 18422  df-srg 18438
This theorem is referenced by:  srgpcomppsc  18466  srgbinomlem4  18475
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