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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 18870 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpplusf.2 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | mndpfo 18690 | . 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 5667 –onto→wfo 6535 ‘cfv 6537 Basecbs 17153 +𝑓cplusf 18570 Mndcmnd 18667 Grpcgrp 18863 |
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-rmo 3370 df-reu 3371 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-fo 6543 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-0g 17396 df-plusf 18572 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 |
This theorem is referenced by: resgrpplusfrn 18880 |
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