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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 18568 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndplusf.2 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
4 | 2, 3 | mgmplusf 18512 | . 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 1542 ∈ wcel 2107 × cxp 5632 ⟶wf 6493 ‘cfv 6497 Basecbs 17088 +𝑓cplusf 18499 Mgmcmgm 18500 Mndcmnd 18561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-plusf 18501 df-mgm 18502 df-sgrp 18551 df-mnd 18562 |
This theorem is referenced by: mndpfo 18584 grpplusf 18767 efmndtmd 23468 submtmd 23471 mhmhmeotmd 32565 |
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