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Mirrors > Home > ILE Home > Th. List > srgcl | GIF version |
Description: Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgcl.b | ⊢ 𝐵 = (Base‘𝑅) |
srgcl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
srgcl | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | srgmgp 13082 | . . . 4 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd) |
3 | 2 | 3ad2ant1 1018 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mnd) |
4 | simp2 998 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | srgcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 1, 5 | mgpbasg 13067 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅))) |
7 | 6 | 3ad2ant1 1018 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘𝑅))) |
8 | 4, 7 | eleqtrd 2256 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘(mulGrp‘𝑅))) |
9 | simp3 999 | . . . 4 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
10 | 9, 7 | eleqtrd 2256 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ (Base‘(mulGrp‘𝑅))) |
11 | eqid 2177 | . . . 4 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | |
12 | eqid 2177 | . . . 4 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
13 | 11, 12 | mndcl 12756 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑋 ∈ (Base‘(mulGrp‘𝑅)) ∧ 𝑌 ∈ (Base‘(mulGrp‘𝑅))) → (𝑋(+g‘(mulGrp‘𝑅))𝑌) ∈ (Base‘(mulGrp‘𝑅))) |
14 | 3, 8, 10, 13 | syl3anc 1238 | . 2 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘(mulGrp‘𝑅))𝑌) ∈ (Base‘(mulGrp‘𝑅))) |
15 | srgcl.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
16 | 1, 15 | mgpplusgg 13065 | . . . 4 ⊢ (𝑅 ∈ SRing → · = (+g‘(mulGrp‘𝑅))) |
17 | 16 | 3ad2ant1 1018 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → · = (+g‘(mulGrp‘𝑅))) |
18 | 17 | oveqd 5889 | . 2 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋(+g‘(mulGrp‘𝑅))𝑌)) |
19 | 14, 18, 7 | 3eltr4d 2261 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 .rcmulr 12529 Mndcmnd 12749 mulGrpcmgp 13061 SRingcsrg 13077 |
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-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-riota 5828 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-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-plusg 12541 df-mulr 12542 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-mgp 13062 df-srg 13078 |
This theorem is referenced by: srgfcl 13087 srgmulgass 13103 srgpcomppsc 13106 srglmhm 13107 srgrmhm 13108 reldvdsrsrg 13192 dvdsrvald 13193 dvdsrd 13194 dvdsrex 13198 |
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