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Theorem srgpcomp 18448
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 6612 . . . . . 6 (𝑥 = 0 → (𝑥 𝐵) = (0 𝐵))
32oveq1d 6620 . . . . 5 (𝑥 = 0 → ((𝑥 𝐵) × 𝐴) = ((0 𝐵) × 𝐴))
42oveq2d 6621 . . . . 5 (𝑥 = 0 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (0 𝐵)))
53, 4eqeq12d 2641 . . . 4 (𝑥 = 0 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵))))
65imbi2d 330 . . 3 (𝑥 = 0 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))))
7 oveq1 6612 . . . . . 6 (𝑥 = 𝑦 → (𝑥 𝐵) = (𝑦 𝐵))
87oveq1d 6620 . . . . 5 (𝑥 = 𝑦 → ((𝑥 𝐵) × 𝐴) = ((𝑦 𝐵) × 𝐴))
97oveq2d 6621 . . . . 5 (𝑥 = 𝑦 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝑦 𝐵)))
108, 9eqeq12d 2641 . . . 4 (𝑥 = 𝑦 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))))
1110imbi2d 330 . . 3 (𝑥 = 𝑦 → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)))))
12 oveq1 6612 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝑥 𝐵) = ((𝑦 + 1) 𝐵))
1312oveq1d 6620 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝑥 𝐵) × 𝐴) = (((𝑦 + 1) 𝐵) × 𝐴))
1412oveq2d 6621 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐴 × (𝑥 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
1513, 14eqeq12d 2641 . . . 4 (𝑥 = (𝑦 + 1) → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
1615imbi2d 330 . . 3 (𝑥 = (𝑦 + 1) → ((𝜑 → ((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵))) ↔ (𝜑 → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))))
17 oveq1 6612 . . . . . 6 (𝑥 = 𝐾 → (𝑥 𝐵) = (𝐾 𝐵))
1817oveq1d 6620 . . . . 5 (𝑥 = 𝐾 → ((𝑥 𝐵) × 𝐴) = ((𝐾 𝐵) × 𝐴))
1917oveq2d 6621 . . . . 5 (𝑥 = 𝐾 → (𝐴 × (𝑥 𝐵)) = (𝐴 × (𝐾 𝐵)))
2018, 19eqeq12d 2641 . . . 4 (𝑥 = 𝐾 → (((𝑥 𝐵) × 𝐴) = (𝐴 × (𝑥 𝐵)) ↔ ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵))))
2120imbi2d 330 . . 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 18411 . . . . . . 7 𝑆 = (Base‘𝐺)
26 eqid 2626 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2723, 26ringidval 18419 . . . . . . 7 (1r𝑅) = (0g𝐺)
28 srgpcomp.e . . . . . . 7 = (.g𝐺)
2925, 27, 28mulg0 17462 . . . . . 6 (𝐵𝑆 → (0 𝐵) = (1r𝑅))
3022, 29syl 17 . . . . 5 (𝜑 → (0 𝐵) = (1r𝑅))
3130oveq1d 6620 . . . 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 18438 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → (𝐴 × (1r𝑅)) = 𝐴)
3632, 33, 35syl2anc 692 . . . . 5 (𝜑 → (𝐴 × (1r𝑅)) = 𝐴)
3730oveq2d 6621 . . . . 5 (𝜑 → (𝐴 × (0 𝐵)) = (𝐴 × (1r𝑅)))
3824, 34, 26srglidm 18437 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝐴𝑆) → ((1r𝑅) × 𝐴) = 𝐴)
3932, 33, 38syl2anc 692 . . . . 5 (𝜑 → ((1r𝑅) × 𝐴) = 𝐴)
4036, 37, 393eqtr4rd 2671 . . . 4 (𝜑 → ((1r𝑅) × 𝐴) = (𝐴 × (0 𝐵)))
4131, 40eqtrd 2660 . . 3 (𝜑 → ((0 𝐵) × 𝐴) = (𝐴 × (0 𝐵)))
4223srgmgp 18426 . . . . . . . . . . . . 13 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
4332, 42syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Mnd)
4443adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐺 ∈ Mnd)
45 simpr 477 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
4622adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐵𝑆)
4723, 34mgpplusg 18409 . . . . . . . . . . . 12 × = (+g𝐺)
4825, 28, 47mulgnn0p1 17468 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝐵𝑆) → ((𝑦 + 1) 𝐵) = ((𝑦 𝐵) × 𝐵))
4944, 45, 46, 48syl3anc 1323 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 + 1) 𝐵) = ((𝑦 𝐵) × 𝐵))
5049oveq1d 6620 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐵) × 𝐴))
51 srgpcomp.c . . . . . . . . . . . . 13 (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))
5251eqcomd 2632 . . . . . . . . . . . 12 (𝜑 → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5352adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝐵 × 𝐴) = (𝐴 × 𝐵))
5453oveq2d 6621 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × (𝐵 × 𝐴)) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
5532adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝑅 ∈ SRing)
5625, 28mulgnn0cl 17474 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0𝐵𝑆) → (𝑦 𝐵) ∈ 𝑆)
5744, 45, 46, 56syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → (𝑦 𝐵) ∈ 𝑆)
5833adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℕ0) → 𝐴𝑆)
5924, 34srgass 18429 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐵𝑆𝐴𝑆)) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
6055, 57, 46, 58, 59syl13anc 1325 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = ((𝑦 𝐵) × (𝐵 × 𝐴)))
6124, 34srgass 18429 . . . . . . . . . . 11 ((𝑅 ∈ SRing ∧ ((𝑦 𝐵) ∈ 𝑆𝐴𝑆𝐵𝑆)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6255, 57, 58, 46, 61syl13anc 1325 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝑦 𝐵) × (𝐴 × 𝐵)))
6354, 60, 623eqtr4d 2670 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6450, 63eqtrd 2660 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
6564adantr 481 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (((𝑦 𝐵) × 𝐴) × 𝐵))
66 oveq1 6612 . . . . . . . 8 (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 𝐵) × 𝐴) × 𝐵) = ((𝐴 × (𝑦 𝐵)) × 𝐵))
6724, 34srgass 18429 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ (𝐴𝑆 ∧ (𝑦 𝐵) ∈ 𝑆𝐵𝑆)) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6855, 58, 57, 46, 67syl13anc 1325 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 𝐵) × 𝐵)))
6949eqcomd 2632 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℕ0) → ((𝑦 𝐵) × 𝐵) = ((𝑦 + 1) 𝐵))
7069oveq2d 6621 . . . . . . . . 9 ((𝜑𝑦 ∈ ℕ0) → (𝐴 × ((𝑦 𝐵) × 𝐵)) = (𝐴 × ((𝑦 + 1) 𝐵)))
7168, 70eqtrd 2660 . . . . . . . 8 ((𝜑𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 𝐵)) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7266, 71sylan9eqr 2682 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 𝐵) × 𝐴) × 𝐵) = (𝐴 × ((𝑦 + 1) 𝐵)))
7365, 72eqtrd 2660 . . . . . 6 (((𝜑𝑦 ∈ ℕ0) ∧ ((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵))) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵)))
7473ex 450 . . . . 5 ((𝜑𝑦 ∈ ℕ0) → (((𝑦 𝐵) × 𝐴) = (𝐴 × (𝑦 𝐵)) → (((𝑦 + 1) 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) 𝐵))))
7574expcom 451 . . . 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 11416 . 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 384   = wceq 1480  wcel 1992  cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  0cn0 11237  Basecbs 15776  .rcmulr 15858  Mndcmnd 17210  .gcmg 17456  mulGrpcmgp 18405  1rcur 18417  SRingcsrg 18421
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-seq 12739  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mulg 17457  df-mgp 18406  df-ur 18418  df-srg 18422
This theorem is referenced by:  srgpcompp  18449  mplcoe5lem  19381
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