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Mirrors > Home > ILE Home > Th. List > rngbaseg | GIF version |
Description: The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
Ref | Expression |
---|---|
rngfn.r | ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
Ref | Expression |
---|---|
rngbaseg | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngfn.r | . . 3 ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
2 | 1 | rngstrg 12063 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
3 | simp1 981 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
4 | basendxnn 12003 | . . . 4 ⊢ (Base‘ndx) ∈ ℕ | |
5 | opexg 4145 | . . . 4 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → 〈(Base‘ndx), 𝐵〉 ∈ V) | |
6 | 4, 3, 5 | sylancr 410 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 〈(Base‘ndx), 𝐵〉 ∈ V) |
7 | tpid1g 3630 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ V → 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉}) | |
8 | 7, 1 | eleqtrrdi 2231 | . . 3 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ V → 〈(Base‘ndx), 𝐵〉 ∈ 𝑅) |
9 | 6, 8 | syl 14 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 〈(Base‘ndx), 𝐵〉 ∈ 𝑅) |
10 | 2, 3, 9 | opelstrbas 12045 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {ctp 3524 〈cop 3525 ‘cfv 5118 1c1 7614 ℕcn 8713 3c3 8765 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: (None) |
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