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Theorem srgpcomp 19279
Description: If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgpcomp.s 𝑆 = (Base‘𝑅)
srgpcomp.m × = (.r𝑅)
srgpcomp.g 𝐺 = (mulGrp‘𝑅)
srgpcomp.e = (.g𝐺)
srgpcomp.r (𝜑𝑅 ∈ SRing)
srgpcomp.a (𝜑𝐴𝑆)
srgpcomp.b (𝜑𝐵𝑆)
srgpcomp.k (𝜑𝐾 ∈ ℕ0)
srgpcomp.c (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))
Assertion
Ref Expression
srgpcomp (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))

Proof of Theorem srgpcomp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgpcomp.k . 2 (𝜑𝐾 ∈ ℕ0)
2 oveq1 7146 . . . . . 6 (𝑥 = 0 → (𝑥 𝐵) = (0 𝐵))
32oveq1d 7154 . . . . 5 (𝑥 = 0 → ((𝑥 𝐵) × 𝐴) = ((0 𝐵) × 𝐴))
42oveq2d 7155 . . . . 5 (𝑥 = 0 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (0 𝐵)))
53, 4eqeq12d 2817 . . . 4 (𝑥 = 0 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵))))
65imbi2d 344 . . 3 (𝑥 = 0 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))))
7 oveq1 7146 . . . . . 6 (𝑥 = 𝑦 → (𝑥 𝐵) = (𝑦 𝐵))
87oveq1d 7154 . . . . 5 (𝑥 = 𝑦 → ((𝑥 𝐵) × 𝐴) = ((𝑦 𝐵) × 𝐴))
97oveq2d 7155 . . . . 5 (𝑥 = 𝑦 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝑦 𝐵)))
108, 9eqeq12d 2817 . . . 4 (𝑥 = 𝑦 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))))
1110imbi2d 344 . . 3 (𝑥 = 𝑦 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)))))
12 oveq1 7146 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑥 𝐵) = ((𝑦 + 1) 𝐵))
1312oveq1d 7154 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑥 𝐵) × 𝐴) = (((𝑦 + 1) 𝐵) × 𝐴))
1412oveq2d 7155 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐴 × (𝑥 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
1513, 14eqeq12d 2817 . . . 4 (𝑥 = (𝑦 + 1) → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
1615imbi2d 344 . . 3 (𝑥 = (𝑦 + 1) → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
17 oveq1 7146 . . . . . 6 (𝑥 = 𝐾 → (𝑥 𝐵) = (𝐾 𝐵))
1817oveq1d 7154 . . . . 5 (𝑥 = 𝐾 → ((𝑥 𝐵) × 𝐴) = ((𝐾 𝐵) × 𝐴))
1917oveq2d 7155 . . . . 5 (𝑥 = 𝐾 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝐾 𝐵)))
2018, 19eqeq12d 2817 . . . 4 (𝑥 = 𝐾 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵))))
2120imbi2d 344 . . 3 (𝑥 = 𝐾 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))))
22 srgpcomp.b . . . . . 6 (𝜑𝐵𝑆)
23 srgpcomp.g . . . . . . . 8 𝐺 = (mulGrp‘𝑅)
24 srgpcomp.s . . . . . . . 8 𝑆 = (Base‘𝑅)
2523, 24mgpbas 19242 . . . . . . 7 𝑆 = (Base‘𝐺)
26 eqid 2801 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2723, 26ringidval 19250 . . . . . . 7 (1r𝑅) = (0g𝐺)
28 srgpcomp.e . . . . . . 7 = (.g𝐺)
2925, 27, 28mulg0 18227 . . . . . 6 (𝐵𝑆 → (0 𝐵) = (1r𝑅))
3022, 29syl 17 . . . . 5 (𝜑 → (0 𝐵) = (1r𝑅))
3130oveq1d 7154 . . . 4 (𝜑 → ((0 𝐵) × 𝐴) = ((1r𝑅) × 𝐴))
32 srgpcomp.r . . . . . 6 (𝜑𝑅 ∈ SRing)
33 srgpcomp.a . . . . . 6 (𝜑𝐴𝑆)
34 srgpcomp.m . . . . . . 7 × = (.r𝑅)
3524, 34, 26srgridm 19269 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → (𝐴 × (1r𝑅)) = 𝐴)
3632, 33, 35syl2anc 587 . . . . 5 (𝜑 → (𝐴 × (1r𝑅)) = 𝐴)
3730oveq2d 7155 . . . . 5 (𝜑 → (𝐴 × (0 𝐵)) = (𝐴 × (1r𝑅)))
3824, 34, 26srglidm 19268 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → ((1r𝑅) × 𝐴) = 𝐴)
3932, 33, 38syl2anc 587 . . . . 5 (𝜑 → ((1r𝑅) × 𝐴) = 𝐴)
4036, 37, 393eqtr4rd 2847 . . . 4 (𝜑 → ((1r𝑅) × 𝐴) = (𝐴 × (0 𝐵)))
4131, 40eqtrd 2836 . . 3 (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))
4223srgmgp 19257 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
4332, 42syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Mnd)
4443adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐺 ∈ Mnd)
45 simpr 488 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
4622adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐵𝑆)
4723, 34mgpplusg 19240 . . . . . . . . . . . 12 × = (+g𝐺)
4825, 28, 47mulgnn0p1 18235 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝐵𝑆) → ((𝑦 + 1) 𝐵) = ((𝑦 𝐵) × 𝐵))
4944, 45, 46, 48syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 + 1) 𝐵) = ((𝑦 𝐵) × 𝐵))
5049oveq1d 7154 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐵) × 𝐴))
51 srgpcomp.c . . . . . . . . . . . . 13 (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))
5251eqcomd 2807 . . . . . . . . . . . 12 (𝜑 → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5352adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5453oveq2d 7155 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × (𝐵 × 𝐴)) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
5532adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑅 ∈ SRing)
5625, 28mulgnn0cl 18240 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝐵𝑆) → (𝑦 𝐵) ∈ 𝑆)
5744, 45, 46, 56syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝑦 𝐵) ∈ 𝑆)
5833adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐴𝑆)
5924, 34srgass 19260 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐵𝑆𝐴𝑆)) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
6055, 57, 46, 58, 59syl13anc 1369 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
6124, 34srgass 19260 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐴𝑆𝐵𝑆)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6255, 57, 58, 46, 61syl13anc 1369 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6354, 60, 623eqtr4d 2846 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6450, 63eqtrd 2836 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6564adantr 484 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
66 oveq1 7146 . . . . . . . 8 (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝐴 × (𝑦 𝐵)) × 𝐵))
6724, 34srgass 19260 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆 ∧ (𝑦 𝐵) ∈ 𝑆𝐵𝑆)) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6855, 58, 57, 46, 67syl13anc 1369 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6949eqcomd 2807 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × 𝐵) = ((𝑦 + 1) 𝐵))
7069oveq2d 7155 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (𝐴 × ((𝑦 𝐵) × 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
7168, 70eqtrd 2836 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7266, 71sylan9eqr 2858 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 𝐵) × 𝐴) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7365, 72eqtrd 2836 . . . . . 6 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))
7473ex 416 . . . . 5 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
7574expcom 417 . . . 4 (𝑦 ∈ ℕ0 → (𝜑 → (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
7675a2d 29 . . 3 (𝑦 ∈ ℕ0 → ((𝜑 → ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (𝜑 → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
776, 11, 16, 21, 41, 76nn0ind 12069 . 2 (𝐾 ∈ ℕ0 → (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵))))
781, 77mpcom 38 1 (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  0cc0 10530  1c1 10531   + caddc 10533  0cn0 11889  Basecbs 16479  .rcmulr 16562  Mndcmnd 17907  .gcmg 18220  mulGrpcmgp 19236  1rcur 19248  SRingcsrg 19252
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-seq 13369  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-plusg 16574  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mulg 18221  df-mgp 19237  df-ur 19249  df-srg 19253
This theorem is referenced by:  srgpcompp  19280  mplcoe5lem  20711
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