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Mirrors > Home > ILE Home > Th. List > mgpscag | GIF version |
Description: The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgpsca.s | ⊢ 𝑆 = (Scalar‘𝑅) |
Ref | Expression |
---|---|
mgpscag | ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (Scalar‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsca.s | . 2 ⊢ 𝑆 = (Scalar‘𝑅) | |
2 | mulrslid 12582 | . . . . 5 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
3 | 2 | slotex 12481 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V) |
4 | scaslid 12603 | . . . . 5 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ) | |
5 | scandxnplusgndx 12605 | . . . . 5 ⊢ (Scalar‘ndx) ≠ (+g‘ndx) | |
6 | plusgslid 12563 | . . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
7 | 6 | simpri 113 | . . . . 5 ⊢ (+g‘ndx) ∈ ℕ |
8 | 4, 5, 7 | setsslnid 12506 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (.r‘𝑅) ∈ V) → (Scalar‘𝑅) = (Scalar‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
9 | 3, 8 | mpdan 421 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (Scalar‘𝑅) = (Scalar‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
10 | mgpbas.1 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
11 | eqid 2177 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 10, 11 | mgpvalg 13064 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉)) |
13 | 12 | fveq2d 5518 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (Scalar‘𝑀) = (Scalar‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉))) |
14 | 9, 13 | eqtr4d 2213 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Scalar‘𝑅) = (Scalar‘𝑀)) |
15 | 1, 14 | eqtrid 2222 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (Scalar‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3595 ‘cfv 5215 (class class class)co 5872 ℕcn 8915 ndxcnx 12451 sSet csts 12452 Slot cslot 12453 +gcplusg 12528 .rcmulr 12529 Scalarcsca 12531 mulGrpcmgp 13061 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 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-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 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-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-5 8977 df-ndx 12457 df-slot 12458 df-sets 12461 df-plusg 12541 df-mulr 12542 df-sca 12544 df-mgp 13062 |
This theorem is referenced by: (None) |
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