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Mirrors > Home > MPE Home > Th. List > pwsmulg | Structured version Visualization version GIF version |
Description: Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
pwsmulg.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsmulg.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsmulg.s | ⊢ ∙ = (.g‘𝑌) |
pwsmulg.t | ⊢ · = (.g‘𝑅) |
Ref | Expression |
---|---|
pwsmulg | ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 763 | . . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑅 ∈ Mnd) | |
2 | simplr 765 | . . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝐼 ∈ 𝑉) | |
3 | simpr3 1188 | . . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝐴 ∈ 𝐼) | |
4 | pwsmulg.y | . . . . 5 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
5 | pwsmulg.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
6 | 4, 5 | pwspjmhm 17982 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅)) |
7 | 1, 2, 3, 6 | syl3anc 1363 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅)) |
8 | simpr1 1186 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑁 ∈ ℕ0) | |
9 | simpr2 1187 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑋 ∈ 𝐵) | |
10 | pwsmulg.s | . . . 4 ⊢ ∙ = (.g‘𝑌) | |
11 | pwsmulg.t | . . . 4 ⊢ · = (.g‘𝑅) | |
12 | 5, 10, 11 | mhmmulg 18206 | . . 3 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋))) |
13 | 7, 8, 9, 12 | syl3anc 1363 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋))) |
14 | 4 | pwsmnd 17934 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Mnd) |
15 | 14 | adantr 481 | . . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑌 ∈ Mnd) |
16 | 5, 10 | mulgnn0cl 18182 | . . . 4 ⊢ ((𝑌 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∙ 𝑋) ∈ 𝐵) |
17 | 15, 8, 9, 16 | syl3anc 1363 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑁 ∙ 𝑋) ∈ 𝐵) |
18 | fveq1 6662 | . . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴)) | |
19 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) | |
20 | fvex 6676 | . . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V | |
21 | 18, 19, 20 | fvmpt 6761 | . . 3 ⊢ ((𝑁 ∙ 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴)) |
22 | 17, 21 | syl 17 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴)) |
23 | fveq1 6662 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴)) | |
24 | fvex 6676 | . . . . 5 ⊢ (𝑋‘𝐴) ∈ V | |
25 | 23, 19, 24 | fvmpt 6761 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴)) |
26 | 9, 25 | syl 17 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴)) |
27 | 26 | oveq2d 7161 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴))) |
28 | 13, 22, 27 | 3eqtr3d 2861 | 1 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℕ0cn0 11885 Basecbs 16471 ↑s cpws 16708 Mndcmnd 17899 MndHom cmhm 17942 .gcmg 18162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-seq 13358 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-mulg 18163 |
This theorem is referenced by: evl1expd 20436 |
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