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Mirrors > Home > MPE Home > Th. List > srgpcompp | Structured version Visualization version GIF version |
Description: If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.) |
Ref | Expression |
---|---|
srgpcomp.s | ⊢ 𝑆 = (Base‘𝑅) |
srgpcomp.m | ⊢ × = (.r‘𝑅) |
srgpcomp.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
srgpcomp.e | ⊢ ↑ = (.g‘𝐺) |
srgpcomp.r | ⊢ (𝜑 → 𝑅 ∈ SRing) |
srgpcomp.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
srgpcomp.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
srgpcomp.k | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
srgpcomp.c | ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) |
srgpcompp.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
srgpcompp | ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgpcomp.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
2 | srgpcomp.g | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | srgpcomp.s | . . . . 5 ⊢ 𝑆 = (Base‘𝑅) | |
4 | 2, 3 | mgpbas 19855 | . . . 4 ⊢ 𝑆 = (Base‘𝐺) |
5 | srgpcomp.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
6 | 2 | srgmgp 19875 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
8 | srgpcompp.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | srgpcomp.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
10 | 4, 5, 7, 8, 9 | mulgnn0cld 18850 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝐴) ∈ 𝑆) |
11 | srgpcomp.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
12 | srgpcomp.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
13 | 4, 5, 7, 11, 12 | mulgnn0cld 18850 | . . 3 ⊢ (𝜑 → (𝐾 ↑ 𝐵) ∈ 𝑆) |
14 | srgpcomp.m | . . . 4 ⊢ × = (.r‘𝑅) | |
15 | 3, 14 | srgass 19878 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
16 | 1, 10, 13, 9, 15 | syl13anc 1372 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
17 | srgpcomp.c | . . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) | |
18 | 3, 14, 2, 5, 1, 9, 12, 11, 17 | srgpcomp 19897 | . . . 4 ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) |
19 | 18 | oveq2d 7367 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
20 | 3, 14 | srgass 19878 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
21 | 1, 10, 9, 13, 20 | syl13anc 1372 | . . 3 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
22 | 19, 21 | eqtr4d 2779 | . 2 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵))) |
23 | 2, 14 | mgpplusg 19853 | . . . . . 6 ⊢ × = (+g‘𝐺) |
24 | 4, 5, 23 | mulgnn0p1 18840 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
25 | 7, 8, 9, 24 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
26 | 25 | eqcomd 2742 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × 𝐴) = ((𝑁 + 1) ↑ 𝐴)) |
27 | 26 | oveq1d 7366 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
28 | 16, 22, 27 | 3eqtrd 2780 | 1 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 1c1 11010 + caddc 11012 ℕ0cn0 12371 Basecbs 17037 .rcmulr 17088 Mndcmnd 18510 .gcmg 18825 mulGrpcmgp 19849 SRingcsrg 19870 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-seq 13861 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-plusg 17100 df-0g 17277 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-mulg 18826 df-mgp 19850 df-ur 19867 df-srg 19871 |
This theorem is referenced by: srgpcomppsc 19899 |
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