Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rngstrg | GIF version |
Description: A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
Ref | Expression |
---|---|
rngfn.r | ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
Ref | Expression |
---|---|
rngstrg | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngfn.r | . 2 ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
2 | 1nn 8724 | . . 3 ⊢ 1 ∈ ℕ | |
3 | basendx 12002 | . . 3 ⊢ (Base‘ndx) = 1 | |
4 | 1lt2 8882 | . . 3 ⊢ 1 < 2 | |
5 | 2nn 8874 | . . 3 ⊢ 2 ∈ ℕ | |
6 | plusgndx 12041 | . . 3 ⊢ (+g‘ndx) = 2 | |
7 | 2lt3 8883 | . . 3 ⊢ 2 < 3 | |
8 | 3nn 8875 | . . 3 ⊢ 3 ∈ ℕ | |
9 | mulrndx 12058 | . . 3 ⊢ (.r‘ndx) = 3 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3g 12040 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
11 | 1, 10 | eqbrtrid 3958 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 {ctp 3524 〈cop 3525 class class class wbr 3924 ‘cfv 5118 1c1 7614 2c2 8764 3c3 8765 Struct cstr 11944 ndxcnx 11945 Basecbs 11948 +gcplusg 12010 .rcmulr 12011 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-tp 3530 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-2 8772 df-3 8773 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-struct 11950 df-ndx 11951 df-slot 11952 df-base 11954 df-plusg 12023 df-mulr 12024 |
This theorem is referenced by: rngbaseg 12064 rngplusgg 12065 rngmulrg 12066 srngstrd 12070 ipsstrd 12089 |
Copyright terms: Public domain | W3C validator |