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Theorem srgpcomp 20170
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 7426 . . . . . 6 (𝑥 = 0 → (𝑥 𝐵) = (0 𝐵))
32oveq1d 7434 . . . . 5 (𝑥 = 0 → ((𝑥 𝐵) × 𝐴) = ((0 𝐵) × 𝐴))
42oveq2d 7435 . . . . 5 (𝑥 = 0 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (0 𝐵)))
53, 4eqeq12d 2741 . . . 4 (𝑥 = 0 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵))))
65imbi2d 339 . . 3 (𝑥 = 0 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))))
7 oveq1 7426 . . . . . 6 (𝑥 = 𝑦 → (𝑥 𝐵) = (𝑦 𝐵))
87oveq1d 7434 . . . . 5 (𝑥 = 𝑦 → ((𝑥 𝐵) × 𝐴) = ((𝑦 𝐵) × 𝐴))
97oveq2d 7435 . . . . 5 (𝑥 = 𝑦 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝑦 𝐵)))
108, 9eqeq12d 2741 . . . 4 (𝑥 = 𝑦 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))))
1110imbi2d 339 . . 3 (𝑥 = 𝑦 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)))))
12 oveq1 7426 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑥 𝐵) = ((𝑦 + 1) 𝐵))
1312oveq1d 7434 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑥 𝐵) × 𝐴) = (((𝑦 + 1) 𝐵) × 𝐴))
1412oveq2d 7435 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐴 × (𝑥 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
1513, 14eqeq12d 2741 . . . 4 (𝑥 = (𝑦 + 1) → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
1615imbi2d 339 . . 3 (𝑥 = (𝑦 + 1) → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
17 oveq1 7426 . . . . . 6 (𝑥 = 𝐾 → (𝑥 𝐵) = (𝐾 𝐵))
1817oveq1d 7434 . . . . 5 (𝑥 = 𝐾 → ((𝑥 𝐵) × 𝐴) = ((𝐾 𝐵) × 𝐴))
1917oveq2d 7435 . . . . 5 (𝑥 = 𝐾 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝐾 𝐵)))
2018, 19eqeq12d 2741 . . . 4 (𝑥 = 𝐾 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵))))
2120imbi2d 339 . . 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 20092 . . . . . . 7 𝑆 = (Base‘𝐺)
26 eqid 2725 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2723, 26ringidval 20135 . . . . . . 7 (1r𝑅) = (0g𝐺)
28 srgpcomp.e . . . . . . 7 = (.g𝐺)
2925, 27, 28mulg0 19038 . . . . . 6 (𝐵𝑆 → (0 𝐵) = (1r𝑅))
3022, 29syl 17 . . . . 5 (𝜑 → (0 𝐵) = (1r𝑅))
3130oveq1d 7434 . . . 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 20155 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → (𝐴 × (1r𝑅)) = 𝐴)
3632, 33, 35syl2anc 582 . . . . 5 (𝜑 → (𝐴 × (1r𝑅)) = 𝐴)
3730oveq2d 7435 . . . . 5 (𝜑 → (𝐴 × (0 𝐵)) = (𝐴 × (1r𝑅)))
3824, 34, 26srglidm 20154 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → ((1r𝑅) × 𝐴) = 𝐴)
3932, 33, 38syl2anc 582 . . . . 5 (𝜑 → ((1r𝑅) × 𝐴) = 𝐴)
4036, 37, 393eqtr4rd 2776 . . . 4 (𝜑 → ((1r𝑅) × 𝐴) = (𝐴 × (0 𝐵)))
4131, 40eqtrd 2765 . . 3 (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))
4223srgmgp 20143 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
4332, 42syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Mnd)
4443adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐺 ∈ Mnd)
45 simpr 483 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
4622adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐵𝑆)
4723, 34mgpplusg 20090 . . . . . . . . . . . 12 × = (+g𝐺)
4825, 28, 47mulgnn0p1 19048 . . . . . . . . . . 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 7434 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐵) × 𝐴))
51 srgpcomp.c . . . . . . . . . . . . 13 (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))
5251eqcomd 2731 . . . . . . . . . . . 12 (𝜑 → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5352adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5453oveq2d 7435 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × (𝐵 × 𝐴)) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
5532adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑅 ∈ SRing)
5625, 28, 44, 45, 46mulgnn0cld 19058 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝑦 𝐵) ∈ 𝑆)
5733adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐴𝑆)
5824, 34srgass 20146 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐵𝑆𝐴𝑆)) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
5955, 56, 46, 57, 58syl13anc 1369 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
6024, 34srgass 20146 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐴𝑆𝐵𝑆)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6155, 56, 57, 46, 60syl13anc 1369 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6254, 59, 613eqtr4d 2775 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6350, 62eqtrd 2765 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6463adantr 479 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
65 oveq1 7426 . . . . . . . 8 (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝐴 × (𝑦 𝐵)) × 𝐵))
6624, 34srgass 20146 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆 ∧ (𝑦 𝐵) ∈ 𝑆𝐵𝑆)) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6755, 57, 56, 46, 66syl13anc 1369 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6849eqcomd 2731 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × 𝐵) = ((𝑦 + 1) 𝐵))
6968oveq2d 7435 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (𝐴 × ((𝑦 𝐵) × 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
7067, 69eqtrd 2765 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7165, 70sylan9eqr 2787 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 𝐵) × 𝐴) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7264, 71eqtrd 2765 . . . . . 6 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))
7372ex 411 . . . . 5 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
7473expcom 412 . . . 4 (𝑦 ∈ ℕ0 → (𝜑 → (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
7574a2d 29 . . 3 (𝑦 ∈ ℕ0 → ((𝜑 → ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (𝜑 → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
766, 11, 16, 21, 41, 75nn0ind 12690 . 2 (𝐾 ∈ ℕ0 → (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵))))
771, 76mpcom 38 1 (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141   + caddc 11143  0cn0 12505  Basecbs 17183  .rcmulr 17237  Mndcmnd 18697  .gcmg 19031  mulGrpcmgp 20086  1rcur 20133  SRingcsrg 20138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-seq 14003  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-plusg 17249  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mulg 19032  df-mgp 20087  df-ur 20134  df-srg 20139
This theorem is referenced by:  srgpcompp  20171  mplcoe5lem  21999
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