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Mirrors > Home > MPE Home > Th. List > mulginvinv | Structured version Visualization version GIF version |
Description: The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
Ref | Expression |
---|---|
mulginvcom.b | ⊢ 𝐵 = (Base‘𝐺) |
mulginvcom.t | ⊢ · = (.g‘𝐺) |
mulginvcom.i | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
mulginvinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulginvcom.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mulginvcom.i | . . . . 5 ⊢ 𝐼 = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18849 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant2 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) |
5 | mulginvcom.t | . . . 4 ⊢ · = (.g‘𝐺) | |
6 | 1, 5, 2 | mulginvcom 18953 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ (𝐼‘𝑋) ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝐼‘(𝑁 · (𝐼‘𝑋)))) |
7 | 4, 6 | syld3an3 1409 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝐼‘(𝑁 · (𝐼‘𝑋)))) |
8 | 1, 2 | grpinvinv 18866 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
9 | 8 | 3adant2 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
10 | 9 | oveq2d 7410 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝑁 · 𝑋)) |
11 | 7, 10 | eqtr3d 2774 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6533 (class class class)co 7394 ℤcz 12542 Basecbs 17128 Grpcgrp 18796 invgcminusg 18797 .gcmg 18924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-seq 13951 df-0g 17371 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-grp 18799 df-minusg 18800 df-mulg 18925 |
This theorem is referenced by: (None) |
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