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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 12071 | . 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 12081 | . 2 ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
8 | 1 | simpri 112 | . . . . 5 ⊢ (.r‘ndx) ∈ ℕ |
9 | opexg 4150 | . . . . 5 ⊢ (((.r‘ndx) ∈ ℕ ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ V) | |
10 | 8, 5, 9 | sylancr 410 | . . . 4 ⊢ (𝜑 → 〈(.r‘ndx), · 〉 ∈ V) |
11 | tpid3g 3638 | . . . 4 ⊢ (〈(.r‘ndx), · 〉 ∈ V → 〈(.r‘ndx), · 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉}) | |
12 | elun1 3243 | . . . 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 2233 | . 2 ⊢ (𝜑 → 〈(.r‘ndx), · 〉 ∈ 𝑅) |
15 | 1, 7, 5, 14 | opelstrsl 12055 | 1 ⊢ (𝜑 → · = (.r‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {csn 3527 {ctp 3529 〈cop 3530 ‘cfv 5123 1c1 7621 ℕcn 8720 4c4 8773 ndxcnx 11956 Slot cslot 11958 Basecbs 11959 +gcplusg 12021 .rcmulr 12022 *𝑟cstv 12023 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-struct 11961 df-ndx 11962 df-slot 11963 df-base 11965 df-plusg 12034 df-mulr 12035 df-starv 12036 |
This theorem is referenced by: (None) |
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