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Mirrors > Home > ILE Home > Th. List > grpstrg | GIF version |
Description: A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
grpfn.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
Ref | Expression |
---|---|
grpstrg | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 2〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfn.g | . 2 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} | |
2 | df-plusg 12034 | . 2 ⊢ +g = Slot 2 | |
3 | 1lt2 8889 | . 2 ⊢ 1 < 2 | |
4 | 2nn 8881 | . 2 ⊢ 2 ∈ ℕ | |
5 | 1, 2, 3, 4 | 2strstrg 12059 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 2〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cpr 3528 〈cop 3530 class class class wbr 3929 ‘cfv 5123 1c1 7621 2c2 8771 Struct cstr 11955 ndxcnx 11956 Basecbs 11959 +gcplusg 12021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-struct 11961 df-ndx 11962 df-slot 11963 df-base 11965 df-plusg 12034 |
This theorem is referenced by: (None) |
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