![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpplusfo | Structured version Visualization version GIF version |
Description: The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.) |
Ref | Expression |
---|---|
grpplusf.1 | ⊢ 𝐵 = (Base‘𝐺) |
grpplusf.2 | ⊢ 𝐹 = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
grpplusfo | ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18911 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpplusf.2 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | mndpfo 18726 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐹:(𝐵 × 𝐵)–onto→𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 × cxp 5680 –onto→wfo 6551 ‘cfv 6553 Basecbs 17189 +𝑓cplusf 18606 Mndcmnd 18703 Grpcgrp 18904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-0g 17432 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 |
This theorem is referenced by: resgrpplusfrn 18921 |
Copyright terms: Public domain | W3C validator |