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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 2177 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
6 | 5 | rngstrg 12587 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
7 | 2, 3, 4, 6 | syl3anc 1238 | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} Struct 〈1, 3〉) |
8 | srngstrd.s | . . . 4 ⊢ (𝜑 → ∗ ∈ 𝑌) | |
9 | 4nn 9080 | . . . . 5 ⊢ 4 ∈ ℕ | |
10 | starvndx 12591 | . . . . 5 ⊢ (*𝑟‘ndx) = 4 | |
11 | 9, 10 | strle1g 12559 | . . . 4 ⊢ ( ∗ ∈ 𝑌 → {〈(*𝑟‘ndx), ∗ 〉} Struct 〈4, 4〉) |
12 | 8, 11 | syl 14 | . . 3 ⊢ (𝜑 → {〈(*𝑟‘ndx), ∗ 〉} Struct 〈4, 4〉) |
13 | 3lt4 9089 | . . . 4 ⊢ 3 < 4 | |
14 | 13 | a1i 9 | . . 3 ⊢ (𝜑 → 3 < 4) |
15 | 7, 12, 14 | strleund 12556 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) Struct 〈1, 4〉) |
16 | 1, 15 | eqbrtrid 4038 | 1 ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 {csn 3592 {ctp 3594 〈cop 3595 class class class wbr 4003 ‘cfv 5216 1c1 7811 < clt 7990 3c3 8969 4c4 8970 Struct cstr 12452 ndxcnx 12453 Basecbs 12456 +gcplusg 12530 .rcmulr 12531 *𝑟cstv 12532 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-tp 3600 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-fz 10007 df-struct 12458 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-mulr 12544 df-starv 12545 |
This theorem is referenced by: srngbased 12599 srngplusgd 12600 srngmulrd 12601 srnginvld 12602 cnfldstr 13348 |
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