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Mirrors > Home > ILE Home > Th. List > srgpcompp | 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 | . . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | 2 | srgmgp 12964 | . . . . . 6 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
4 | 1, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
5 | srgpcompp.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | srgpcomp.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
7 | srgpcomp.s | . . . . . . . 8 ⊢ 𝑆 = (Base‘𝑅) | |
8 | 2, 7 | mgpbasg 12950 | . . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑆 = (Base‘𝐺)) |
9 | 1, 8 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (Base‘𝐺)) |
10 | 6, 9 | eleqtrd 2256 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐺)) |
11 | eqid 2177 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
12 | srgpcomp.e | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
13 | 11, 12 | mulgnn0cl 12875 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ (Base‘𝐺)) → (𝑁 ↑ 𝐴) ∈ (Base‘𝐺)) |
14 | 4, 5, 10, 13 | syl3anc 1238 | . . . 4 ⊢ (𝜑 → (𝑁 ↑ 𝐴) ∈ (Base‘𝐺)) |
15 | 14, 9 | eleqtrrd 2257 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝐴) ∈ 𝑆) |
16 | srgpcomp.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
17 | srgpcomp.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
18 | 17, 9 | eleqtrd 2256 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐺)) |
19 | 11, 12 | mulgnn0cl 12875 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐾 ↑ 𝐵) ∈ (Base‘𝐺)) |
20 | 4, 16, 18, 19 | syl3anc 1238 | . . . 4 ⊢ (𝜑 → (𝐾 ↑ 𝐵) ∈ (Base‘𝐺)) |
21 | 20, 9 | eleqtrrd 2257 | . . 3 ⊢ (𝜑 → (𝐾 ↑ 𝐵) ∈ 𝑆) |
22 | srgpcomp.m | . . . 4 ⊢ × = (.r‘𝑅) | |
23 | 7, 22 | srgass 12967 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
24 | 1, 15, 21, 6, 23 | syl13anc 1240 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴))) |
25 | srgpcomp.c | . . . . 5 ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) | |
26 | 7, 22, 2, 12, 1, 6, 17, 16, 25 | srgpcomp 12986 | . . . 4 ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) |
27 | 26 | oveq2d 5884 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
28 | 7, 22 | srgass 12967 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ ((𝑁 ↑ 𝐴) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ (𝐾 ↑ 𝐵) ∈ 𝑆)) → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
29 | 1, 15, 6, 21, 28 | syl13anc 1240 | . . 3 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
30 | 27, 29 | eqtr4d 2213 | . 2 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵))) |
31 | 2, 22 | mgpplusgg 12948 | . . . . . 6 ⊢ (𝑅 ∈ SRing → × = (+g‘𝐺)) |
32 | 1, 31 | syl 14 | . . . . 5 ⊢ (𝜑 → × = (+g‘𝐺)) |
33 | 32 | oveqd 5885 | . . . 4 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × 𝐴) = ((𝑁 ↑ 𝐴)(+g‘𝐺)𝐴)) |
34 | eqid 2177 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
35 | 11, 12, 34 | mulgnn0p1 12870 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴)(+g‘𝐺)𝐴)) |
36 | 4, 5, 10, 35 | syl3anc 1238 | . . . 4 ⊢ (𝜑 → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴)(+g‘𝐺)𝐴)) |
37 | 33, 36 | eqtr4d 2213 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × 𝐴) = ((𝑁 + 1) ↑ 𝐴)) |
38 | 37 | oveq1d 5883 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
39 | 24, 30, 38 | 3eqtrd 2214 | 1 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 1c1 7790 + caddc 7792 ℕ0cn0 9152 Basecbs 12432 +gcplusg 12505 .rcmulr 12506 Mndcmnd 12696 .gcmg 12859 mulGrpcmgp 12944 SRingcsrg 12959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-3 8955 df-n0 9153 df-z 9230 df-uz 9505 df-seqfrec 10419 df-ndx 12435 df-slot 12436 df-base 12438 df-sets 12439 df-plusg 12518 df-mulr 12519 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-minusg 12758 df-mulg 12860 df-mgp 12945 df-ur 12956 df-srg 12960 |
This theorem is referenced by: srgpcomppsc 12988 |
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