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Mirrors > Home > MPE Home > Th. List > srgfcl | Structured version Visualization version GIF version |
Description: Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.) |
Ref | Expression |
---|---|
srgfcl.b | ⊢ 𝐵 = (Base‘𝑅) |
srgfcl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
srgfcl | ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . 2 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · Fn (𝐵 × 𝐵)) | |
2 | srgfcl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srgfcl.t | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
4 | 2, 3 | srgcl 18512 | . . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵) |
5 | 4 | 3expb 1266 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵) |
6 | 5 | ralrimivva 2971 | . . . . 5 ⊢ (𝑅 ∈ SRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
7 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → ( · ‘𝑐) = ( · ‘〈𝑎, 𝑏〉)) | |
8 | 7 | eleq1d 2686 | . . . . . . 7 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ ( · ‘〈𝑎, 𝑏〉) ∈ 𝐵)) |
9 | df-ov 6653 | . . . . . . . . 9 ⊢ (𝑎 · 𝑏) = ( · ‘〈𝑎, 𝑏〉) | |
10 | 9 | eqcomi 2631 | . . . . . . . 8 ⊢ ( · ‘〈𝑎, 𝑏〉) = (𝑎 · 𝑏) |
11 | 10 | eleq1i 2692 | . . . . . . 7 ⊢ (( · ‘〈𝑎, 𝑏〉) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵) |
12 | 8, 11 | syl6bb 276 | . . . . . 6 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵)) |
13 | 12 | ralxp 5263 | . . . . 5 ⊢ (∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
14 | 6, 13 | sylibr 224 | . . . 4 ⊢ (𝑅 ∈ SRing → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
15 | 14 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
16 | fnfvrnss 6390 | . . 3 ⊢ (( · Fn (𝐵 × 𝐵) ∧ ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) → ran · ⊆ 𝐵) | |
17 | 1, 15, 16 | syl2anc 693 | . 2 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ran · ⊆ 𝐵) |
18 | df-f 5892 | . 2 ⊢ ( · :(𝐵 × 𝐵)⟶𝐵 ↔ ( · Fn (𝐵 × 𝐵) ∧ ran · ⊆ 𝐵)) | |
19 | 1, 17, 18 | sylanbrc 698 | 1 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 〈cop 4183 × cxp 5112 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 SRingcsrg 18505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mgp 18490 df-srg 18506 |
This theorem is referenced by: (None) |
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