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Theorem srgmulgass 12985
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 5875 . . . . . . . 8 (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋))
21oveq1d 5883 . . . . . . 7 (𝑥 = 0 → ((𝑥 · 𝑋) × 𝑌) = ((0 · 𝑋) × 𝑌))
3 oveq1 5875 . . . . . . 7 (𝑥 = 0 → (𝑥 · (𝑋 × 𝑌)) = (0 · (𝑋 × 𝑌)))
42, 3eqeq12d 2192 . . . . . 6 (𝑥 = 0 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌))))
54imbi2d 230 . . . . 5 (𝑥 = 0 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))))
6 oveq1 5875 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋))
76oveq1d 5883 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 · 𝑋) × 𝑌) = ((𝑦 · 𝑋) × 𝑌))
8 oveq1 5875 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 · (𝑋 × 𝑌)) = (𝑦 · (𝑋 × 𝑌)))
97, 8eqeq12d 2192 . . . . . 6 (𝑥 = 𝑦 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))))
109imbi2d 230 . . . . 5 (𝑥 = 𝑦 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)))))
11 oveq1 5875 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋))
1211oveq1d 5883 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) × 𝑌) = (((𝑦 + 1) · 𝑋) × 𝑌))
13 oveq1 5875 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑥 · (𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
1412, 13eqeq12d 2192 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌))))
1514imbi2d 230 . . . . 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 5875 . . . . . . . 8 (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋))
1716oveq1d 5883 . . . . . . 7 (𝑥 = 𝑁 → ((𝑥 · 𝑋) × 𝑌) = ((𝑁 · 𝑋) × 𝑌))
18 oveq1 5875 . . . . . . 7 (𝑥 = 𝑁 → (𝑥 · (𝑋 × 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
1917, 18eqeq12d 2192 . . . . . 6 (𝑥 = 𝑁 → (((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌)) ↔ ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
2019imbi2d 230 . . . . 5 (𝑥 = 𝑁 → ((((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑥 · 𝑋) × 𝑌) = (𝑥 · (𝑋 × 𝑌))) ↔ (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
21 simpr 110 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ SRing)
22 simpr 110 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
2322adantr 276 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑌𝐵)
24 srgmulgass.b . . . . . . . 8 𝐵 = (Base‘𝑅)
25 srgmulgass.t . . . . . . . 8 × = (.r𝑅)
26 eqid 2177 . . . . . . . 8 (0g𝑅) = (0g𝑅)
2724, 25, 26srglz 12981 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑌𝐵) → ((0g𝑅) × 𝑌) = (0g𝑅))
2821, 23, 27syl2anc 411 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0g𝑅) × 𝑌) = (0g𝑅))
29 simpl 109 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3029adantr 276 . . . . . . . 8 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑋𝐵)
31 srgmulgass.m . . . . . . . . 9 · = (.g𝑅)
3224, 26, 31mulg0 12864 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = (0g𝑅))
3330, 32syl 14 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · 𝑋) = (0g𝑅))
3433oveq1d 5883 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = ((0g𝑅) × 𝑌))
3524, 25srgcl 12966 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 × 𝑌) ∈ 𝐵)
3621, 30, 23, 35syl3anc 1238 . . . . . . 7 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
3724, 26, 31mulg0 12864 . . . . . . 7 ((𝑋 × 𝑌) ∈ 𝐵 → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3836, 37syl 14 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (0 · (𝑋 × 𝑌)) = (0g𝑅))
3928, 34, 383eqtr4d 2220 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((0 · 𝑋) × 𝑌) = (0 · (𝑋 × 𝑌)))
40 srgmnd 12963 . . . . . . . . . . . . . 14 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
4140adantl 277 . . . . . . . . . . . . 13 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → 𝑅 ∈ Mnd)
4241adantl 277 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ Mnd)
43 simpl 109 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑦 ∈ ℕ0)
4430adantl 277 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑋𝐵)
45 eqid 2177 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
4624, 31, 45mulgnn0p1 12870 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4742, 43, 44, 46syl3anc 1238 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g𝑅)𝑋))
4847oveq1d 5883 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌))
4921adantl 277 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑅 ∈ SRing)
5024, 31mulgnn0cl 12875 . . . . . . . . . . . 12 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝑋𝐵) → (𝑦 · 𝑋) ∈ 𝐵)
5142, 43, 44, 50syl3anc 1238 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑦 · 𝑋) ∈ 𝐵)
5223adantl 277 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → 𝑌𝐵)
5324, 45, 25srgdir 12971 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 · 𝑋) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5449, 51, 44, 52, 53syl13anc 1240 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 · 𝑋)(+g𝑅)𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5548, 54eqtrd 2210 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
5655adantr 276 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)))
57 oveq1 5875 . . . . . . . . 9 (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
58353expb 1204 . . . . . . . . . . . . 13 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 × 𝑌) ∈ 𝐵)
5958ancoms 268 . . . . . . . . . . . 12 (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (𝑋 × 𝑌) ∈ 𝐵)
6059adantl 277 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → (𝑋 × 𝑌) ∈ 𝐵)
6124, 31, 45mulgnn0p1 12870 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ (𝑋 × 𝑌) ∈ 𝐵) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6242, 43, 60, 61syl3anc 1238 . . . . . . . . . 10 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 + 1) · (𝑋 × 𝑌)) = ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)))
6362eqcomd 2183 . . . . . . . . 9 ((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) → ((𝑦 · (𝑋 × 𝑌))(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6457, 63sylan9eqr 2232 . . . . . . . 8 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 · 𝑋) × 𝑌)(+g𝑅)(𝑋 × 𝑌)) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6556, 64eqtrd 2210 . . . . . . 7 (((𝑦 ∈ ℕ0 ∧ ((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing)) ∧ ((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌))) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))
6665exp31 364 . . . . . 6 (𝑦 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → (((𝑦 · 𝑋) × 𝑌) = (𝑦 · (𝑋 × 𝑌)) → (((𝑦 + 1) · 𝑋) × 𝑌) = ((𝑦 + 1) · (𝑋 × 𝑌)))))
6766a2d 26 . . . . 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 9343 . . . 4 (𝑁 ∈ ℕ0 → (((𝑋𝐵𝑌𝐵) ∧ 𝑅 ∈ SRing) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
6968expd 258 . . 3 (𝑁 ∈ ℕ0 → ((𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))))
70693impib 1201 . 2 ((𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵) → (𝑅 ∈ SRing → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))))
7170impcom 125 1 ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  0cc0 7789  1c1 7790   + caddc 7792  0cn0 9152  Basecbs 12432  +gcplusg 12505  .rcmulr 12506  0gc0g 12640  Mndcmnd 12696  .gcmg 12859  SRingcsrg 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-3 8955  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-minusg 12758  df-mulg 12860  df-cmn 12904  df-mgp 12945  df-srg 12960
This theorem is referenced by:  srgpcomppsc  12988
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