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Mirrors > Home > ILE Home > Th. List > srngmulrd | GIF version |
Description: The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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 |
---|---|
srngmulrd | ⊢ (𝜑 → · = (.r‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulrslid 12556 | . 2 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
2 | srngstr.r | . . 3 ⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) | |
3 | srngstrd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | srngstrd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
5 | srngstrd.m | . . 3 ⊢ (𝜑 → · ∈ 𝑋) | |
6 | srngstrd.s | . . 3 ⊢ (𝜑 → ∗ ∈ 𝑌) | |
7 | 2, 3, 4, 5, 6 | srngstrd 12566 | . 2 ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
8 | 1 | simpri 113 | . . . . 5 ⊢ (.r‘ndx) ∈ ℕ |
9 | opexg 4224 | . . . . 5 ⊢ (((.r‘ndx) ∈ ℕ ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ V) | |
10 | 8, 5, 9 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(.r‘ndx), · 〉 ∈ V) |
11 | tpid3g 3706 | . . . 4 ⊢ (〈(.r‘ndx), · 〉 ∈ V → 〈(.r‘ndx), · 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉}) | |
12 | elun1 3302 | . . . 4 ⊢ (〈(.r‘ndx), · 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} → 〈(.r‘ndx), · 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉})) | |
13 | 10, 11, 12 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(.r‘ndx), · 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉})) |
14 | 13, 2 | eleqtrrdi 2271 | . 2 ⊢ (𝜑 → 〈(.r‘ndx), · 〉 ∈ 𝑅) |
15 | 1, 7, 5, 14 | opelstrsl 12539 | 1 ⊢ (𝜑 → · = (.r‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 {csn 3591 {ctp 3593 〈cop 3594 ‘cfv 5211 1c1 7790 ℕcn 8895 4c4 8948 ndxcnx 12429 Slot cslot 12431 Basecbs 12432 +gcplusg 12505 .rcmulr 12506 *𝑟cstv 12507 |
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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 |
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 3576 df-sn 3597 df-pr 3598 df-tp 3599 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-n0 9153 df-z 9230 df-uz 9505 df-fz 9983 df-struct 12434 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-mulr 12519 df-starv 12520 |
This theorem is referenced by: (None) |
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