![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mndplusf | Structured version Visualization version GIF version |
Description: The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.) |
Ref | Expression |
---|---|
mndplusf.1 | ⊢ 𝐵 = (Base‘𝐺) |
mndplusf.2 | ⊢ ⨣ = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
mndplusf | ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndmgm 18631 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndplusf.2 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
4 | 2, 3 | mgmplusf 18570 | . 2 ⊢ (𝐺 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 × cxp 5674 ⟶wf 6539 ‘cfv 6543 Basecbs 17143 +𝑓cplusf 18557 Mgmcmgm 18558 Mndcmnd 18624 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-plusf 18559 df-mgm 18560 df-sgrp 18609 df-mnd 18625 |
This theorem is referenced by: mndpfo 18647 grpplusf 18833 efmndtmd 23604 submtmd 23607 mhmhmeotmd 32902 |
Copyright terms: Public domain | W3C validator |