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Mirrors > Home > ILE Home > Th. List > grpplusf | GIF version |
Description: The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
grpplusf.1 | ⊢ 𝐵 = (Base‘𝐺) |
grpplusf.2 | ⊢ 𝐹 = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
grpplusf | ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 12761 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpplusf.2 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | mndplusf 12713 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐹:(𝐵 × 𝐵)⟶𝐵) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 × cxp 4620 ⟶wf 5207 ‘cfv 5211 Basecbs 12432 +𝑓cplusf 12651 Mndcmnd 12696 Grpcgrp 12754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-plusf 12653 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 |
This theorem is referenced by: (None) |
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