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Mirrors > Home > MPE Home > Th. List > fnmgp | Structured version Visualization version GIF version |
Description: The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
fnmgp | ⊢ mulGrp Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7189 | . 2 ⊢ (𝑥 sSet 〈(+g‘ndx), (.r‘𝑥)〉) ∈ V | |
2 | df-mgp 19240 | . 2 ⊢ mulGrp = (𝑥 ∈ V ↦ (𝑥 sSet 〈(+g‘ndx), (.r‘𝑥)〉)) | |
3 | 1, 2 | fnmpti 6491 | 1 ⊢ mulGrp Fn V |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3494 〈cop 4573 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 sSet csts 16481 +gcplusg 16565 .rcmulr 16566 mulGrpcmgp 19239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-ov 7159 df-mgp 19240 |
This theorem is referenced by: ringidval 19253 mgpf 19309 prdsmgp 19360 prdscrngd 19363 pws1 19366 pwsmgp 19368 |
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