![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pwsexpg | Structured version Visualization version GIF version |
Description: Value of a group exponentiation in a structure power. Compare pwsmulg 19150. (Contributed by SN, 30-Jul-2024.) |
Ref | Expression |
---|---|
pwsexpg.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsexpg.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsexpg.m | ⊢ 𝑀 = (mulGrp‘𝑌) |
pwsexpg.t | ⊢ 𝑇 = (mulGrp‘𝑅) |
pwsexpg.s | ⊢ ∙ = (.g‘𝑀) |
pwsexpg.g | ⊢ · = (.g‘𝑇) |
pwsexpg.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
pwsexpg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
pwsexpg.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
pwsexpg.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
pwsexpg.a | ⊢ (𝜑 → 𝐴 ∈ 𝐼) |
Ref | Expression |
---|---|
pwsexpg | ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsexpg.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
2 | pwsexpg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
3 | pwsexpg.m | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑌) | |
4 | pwsexpg.t | . . . 4 ⊢ 𝑇 = (mulGrp‘𝑅) | |
5 | pwsexpg.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | pwsexpg.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | pwsexpg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐼) | |
8 | 1, 2, 3, 4, 5, 6, 7 | pwspjmhmmgpd 20342 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇)) |
9 | pwsexpg.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
10 | pwsexpg.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 3, 2 | mgpbas 20158 | . . . 4 ⊢ 𝐵 = (Base‘𝑀) |
12 | pwsexpg.s | . . . 4 ⊢ ∙ = (.g‘𝑀) | |
13 | pwsexpg.g | . . . 4 ⊢ · = (.g‘𝑇) | |
14 | 11, 12, 13 | mhmmulg 19146 | . . 3 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋))) |
15 | 8, 9, 10, 14 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋))) |
16 | 1 | pwsring 20338 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring) |
17 | 5, 6, 16 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring) |
18 | 3 | ringmgp 20257 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑀 ∈ Mnd) |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
20 | 11, 12, 19, 9, 10 | mulgnn0cld 19126 | . . 3 ⊢ (𝜑 → (𝑁 ∙ 𝑋) ∈ 𝐵) |
21 | fveq1 6906 | . . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴)) | |
22 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) | |
23 | fvex 6920 | . . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V | |
24 | 21, 22, 23 | fvmpt 7016 | . . 3 ⊢ ((𝑁 ∙ 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴)) |
25 | 20, 24 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴)) |
26 | fveq1 6906 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴)) | |
27 | fvex 6920 | . . . . 5 ⊢ (𝑋‘𝐴) ∈ V | |
28 | 26, 22, 27 | fvmpt 7016 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴)) |
29 | 10, 28 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴)) |
30 | 29 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴))) |
31 | 15, 25, 30 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℕ0cn0 12524 Basecbs 17245 ↑s cpws 17493 Mndcmnd 18760 MndHom cmhm 18807 .gcmg 19098 mulGrpcmgp 20152 Ringcrg 20251 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-seq 14040 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-mulg 19099 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 |
This theorem is referenced by: evls1expd 22387 evlsvvval 42550 evlsexpval 42554 |
Copyright terms: Public domain | W3C validator |