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Mirrors > Home > ILE Home > Th. List > rngmulrg | GIF version |
Description: The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
rngfn.r | ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
Ref | Expression |
---|---|
rngmulrg | ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · = (.r‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulrslid 12530 | . 2 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
2 | rngfn.r | . . 3 ⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | |
3 | 2 | rngstrg 12533 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
4 | simp3 994 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · ∈ 𝑋) | |
5 | 1 | simpri 112 | . . . . 5 ⊢ (.r‘ndx) ∈ ℕ |
6 | opexg 4213 | . . . . 5 ⊢ (((.r‘ndx) ∈ ℕ ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ V) | |
7 | 5, 4, 6 | sylancr 412 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ V) |
8 | tpid3g 3698 | . . . 4 ⊢ (〈(.r‘ndx), · 〉 ∈ V → 〈(.r‘ndx), · 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉}) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉}) |
10 | 9, 2 | eleqtrrdi 2264 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 〈(.r‘ndx), · 〉 ∈ 𝑅) |
11 | 1, 3, 4, 10 | opelstrsl 12514 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · = (.r‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 Vcvv 2730 {ctp 3585 〈cop 3586 ‘cfv 5198 1c1 7775 ℕcn 8878 3c3 8930 ndxcnx 12413 Slot cslot 12415 Basecbs 12416 +gcplusg 12480 .rcmulr 12481 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-tp 3591 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-3 8938 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-struct 12418 df-ndx 12419 df-slot 12420 df-base 12422 df-plusg 12493 df-mulr 12494 |
This theorem is referenced by: (None) |
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