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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 | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | 2 | srgmgp 19254 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
5 | srgpcompp.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | srgpcomp.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
7 | srgpcomp.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝑅) | |
8 | 2, 7 | mgpbas 19239 | . . . . 5 ⊢ 𝑆 = (Base‘𝐺) |
9 | srgpcomp.e | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
10 | 8, 9 | mulgnn0cl 18238 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → (𝑁 ↑ 𝐴) ∈ 𝑆) |
11 | 4, 5, 6, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝐴) ∈ 𝑆) |
12 | srgpcomp.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
13 | srgpcomp.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
14 | 8, 9 | mulgnn0cl 18238 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆) → (𝐾 ↑ 𝐵) ∈ 𝑆) |
15 | 4, 12, 13, 14 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐾 ↑ 𝐵) ∈ 𝑆) |
16 | srgpcomp.m | . . . 4 ⊢ × = (.r‘𝑅) | |
17 | 7, 16 | srgass 19257 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
18 | 1, 11, 15, 6, 17 | syl13anc 1368 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
19 | srgpcomp.c | . . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) | |
20 | 7, 16, 2, 9, 1, 6, 13, 12, 19 | srgpcomp 19276 | . . . 4 ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) |
21 | 20 | oveq2d 7166 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
22 | 7, 16 | srgass 19257 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
23 | 1, 11, 6, 15, 22 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
24 | 21, 23 | eqtr4d 2859 | . 2 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵))) |
25 | 2, 16 | mgpplusg 19237 | . . . . . 6 ⊢ × = (+g‘𝐺) |
26 | 8, 9, 25 | mulgnn0p1 18233 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
27 | 4, 5, 6, 26 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
28 | 27 | eqcomd 2827 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × 𝐴) = ((𝑁 + 1) ↑ 𝐴)) |
29 | 28 | oveq1d 7165 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
30 | 18, 24, 29 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 1c1 10532 + caddc 10534 ℕ0cn0 11891 Basecbs 16477 .rcmulr 16560 Mndcmnd 17905 .gcmg 18218 mulGrpcmgp 19233 SRingcsrg 19249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mulg 18219 df-mgp 19234 df-ur 19246 df-srg 19250 |
This theorem is referenced by: srgpcomppsc 19278 |
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