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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 18333 | . . . . 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 18318 | . . . . 5 ⊢ 𝑆 = (Base‘𝐺) |
9 | srgpcomp.e | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
10 | 8, 9 | mulgnn0cl 17381 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → (𝑁 ↑ 𝐴) ∈ 𝑆) |
11 | 4, 5, 6, 10 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝐴) ∈ 𝑆) |
12 | srgpcomp.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
13 | srgpcomp.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
14 | 8, 9 | mulgnn0cl 17381 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆) → (𝐾 ↑ 𝐵) ∈ 𝑆) |
15 | 4, 12, 13, 14 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐾 ↑ 𝐵) ∈ 𝑆) |
16 | srgpcomp.m | . . . 4 ⊢ × = (.r‘𝑅) | |
17 | 7, 16 | srgass 18336 | . . 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 1320 | . 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 18355 | . . . 4 ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) |
21 | 20 | oveq2d 6565 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
22 | 7, 16 | srgass 18336 | . . . 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 1320 | . . 3 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = ((𝑁 ↑ 𝐴) × (𝐴 × (𝐾 ↑ 𝐵)))) |
24 | 21, 23 | eqtr4d 2647 | . 2 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × ((𝐾 ↑ 𝐵) × 𝐴)) = (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵))) |
25 | 2, 16 | mgpplusg 18316 | . . . . . 6 ⊢ × = (+g‘𝐺) |
26 | 8, 9, 25 | mulgnn0p1 17375 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
27 | 4, 5, 6, 26 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → ((𝑁 + 1) ↑ 𝐴) = ((𝑁 ↑ 𝐴) × 𝐴)) |
28 | 27 | eqcomd 2616 | . . 3 ⊢ (𝜑 → ((𝑁 ↑ 𝐴) × 𝐴) = ((𝑁 + 1) ↑ 𝐴)) |
29 | 28 | oveq1d 6564 | . 2 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × 𝐴) × (𝐾 ↑ 𝐵)) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
30 | 18, 24, 29 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1c1 9816 + caddc 9818 ℕ0cn0 11169 Basecbs 15695 .rcmulr 15769 Mndcmnd 17117 .gcmg 17363 mulGrpcmgp 18312 SRingcsrg 18328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 df-mgp 18313 df-ur 18325 df-srg 18329 |
This theorem is referenced by: srgpcomppsc 18357 |
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