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Mirrors > Home > ILE Home > Th. List > srngstrd | GIF version |
Description: A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
Ref | Expression |
---|---|
srngstr.r | ⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) |
srngstrd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
srngstrd.p | ⊢ (𝜑 → + ∈ 𝑊) |
srngstrd.m | ⊢ (𝜑 → · ∈ 𝑋) |
srngstrd.s | ⊢ (𝜑 → ∗ ∈ 𝑌) |
Ref | Expression |
---|---|
srngstrd | ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngstr.r | . 2 ⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) | |
2 | srngstrd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | srngstrd.p | . . . 4 ⊢ (𝜑 → + ∈ 𝑊) | |
4 | srngstrd.m | . . . 4 ⊢ (𝜑 → · ∈ 𝑋) | |
5 | eqid 2139 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
6 | 5 | rngstrg 12079 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
7 | 2, 3, 4, 6 | syl3anc 1216 | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
8 | srngstrd.s | . . . 4 ⊢ (𝜑 → ∗ ∈ 𝑌) | |
9 | 4nn 8888 | . . . . 5 ⊢ 4 ∈ ℕ | |
10 | starvndx 12083 | . . . . 5 ⊢ (*𝑟‘ndx) = 4 | |
11 | 9, 10 | strle1g 12054 | . . . 4 ⊢ ( ∗ ∈ 𝑌 → {〈(*𝑟‘ndx), ∗ 〉} Struct 〈4, 4〉) |
12 | 8, 11 | syl 14 | . . 3 ⊢ (𝜑 → {〈(*𝑟‘ndx), ∗ 〉} Struct 〈4, 4〉) |
13 | 3lt4 8897 | . . . 4 ⊢ 3 < 4 | |
14 | 13 | a1i 9 | . . 3 ⊢ (𝜑 → 3 < 4) |
15 | 7, 12, 14 | strleund 12052 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) Struct 〈1, 4〉) |
16 | 1, 15 | eqbrtrid 3963 | 1 ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∪ cun 3069 {csn 3527 {ctp 3529 〈cop 3530 class class class wbr 3929 ‘cfv 5123 1c1 7626 < clt 7805 3c3 8777 4c4 8778 Struct cstr 11960 ndxcnx 11961 Basecbs 11964 +gcplusg 12026 .rcmulr 12027 *𝑟cstv 12028 |
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 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-addass 7727 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-0id 7733 ax-rnegex 7734 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 |
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-tp 3535 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 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-n0 8983 df-z 9060 df-uz 9332 df-fz 9796 df-struct 11966 df-ndx 11967 df-slot 11968 df-base 11970 df-plusg 12039 df-mulr 12040 df-starv 12041 |
This theorem is referenced by: srngbased 12087 srngplusgd 12088 srngmulrd 12089 srnginvld 12090 |
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