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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 17780 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndplusf.2 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
4 | 2, 3 | mgmplusf 17731 | . 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 1507 ∈ wcel 2050 × cxp 5401 ⟶wf 6181 ‘cfv 6185 Basecbs 16337 +𝑓cplusf 17719 Mgmcmgm 17720 Mndcmnd 17774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-plusf 17721 df-mgm 17722 df-sgrp 17764 df-mnd 17775 |
This theorem is referenced by: mndpfo 17794 grpplusf 17915 submtmd 22428 mhmhmeotmd 30843 |
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