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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 18674 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndplusf.2 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
4 | 2, 3 | mgmplusf 18583 | . 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 1533 ∈ wcel 2098 × cxp 5667 ⟶wf 6533 ‘cfv 6537 Basecbs 17153 +𝑓cplusf 18570 Mgmcmgm 18571 Mndcmnd 18667 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-plusf 18572 df-mgm 18573 df-sgrp 18652 df-mnd 18668 |
This theorem is referenced by: mndpfo 18690 grpplusf 18878 efmndtmd 23960 submtmd 23963 mhmhmeotmd 33437 |
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