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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 8995 | . . 3 ⊢ 1 ∈ ℕ | |
3 | basendx 12676 | . . 3 ⊢ (Base‘ndx) = 1 | |
4 | 1lt2 9154 | . . 3 ⊢ 1 < 2 | |
5 | 2nn 9146 | . . 3 ⊢ 2 ∈ ℕ | |
6 | plusgndx 12730 | . . 3 ⊢ (+g‘ndx) = 2 | |
7 | 2lt3 9155 | . . 3 ⊢ 2 < 3 | |
8 | 3nn 9147 | . . 3 ⊢ 3 ∈ ℕ | |
9 | mulrndx 12750 | . . 3 ⊢ (.r‘ndx) = 3 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3g 12729 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
11 | 1, 10 | eqbrtrid 4065 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 {ctp 3621 〈cop 3622 class class class wbr 4030 ‘cfv 5255 1c1 7875 2c2 9035 3c3 9036 Struct cstr 12617 ndxcnx 12618 Basecbs 12621 +gcplusg 12698 .rcmulr 12699 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-tp 3627 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-inn 8985 df-2 9043 df-3 9044 df-n0 9244 df-z 9321 df-uz 9596 df-fz 10078 df-struct 12623 df-ndx 12624 df-slot 12625 df-base 12627 df-plusg 12711 df-mulr 12712 |
This theorem is referenced by: rngbaseg 12756 rngplusgg 12757 rngmulrg 12758 srngstrd 12766 ipsstrd 12796 psrvalstrd 14165 |
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