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Mirrors > Home > MPE Home > Th. List > grpplusf | Structured version Visualization version 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 17868 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpplusf.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpplusf.2 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | mndplusf 17748 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐹:(𝐵 × 𝐵)⟶𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 × cxp 5441 ⟶wf 6221 ‘cfv 6225 Basecbs 16312 +𝑓cplusf 17678 Mndcmnd 17733 Grpcgrp 17861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-plusf 17680 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 |
This theorem is referenced by: symgtgp 22393 |
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