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Mirrors > Home > ILE Home > Th. List > srgfcl | 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 110 | . 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 13106 | . . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵) |
5 | 4 | 3expb 1204 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵) |
6 | 5 | ralrimivva 2559 | . . . . 5 ⊢ (𝑅 ∈ SRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
7 | fveq2 5515 | . . . . . . . 8 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → ( · ‘𝑐) = ( · ‘〈𝑎, 𝑏〉)) | |
8 | 7 | eleq1d 2246 | . . . . . . 7 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ ( · ‘〈𝑎, 𝑏〉) ∈ 𝐵)) |
9 | df-ov 5877 | . . . . . . . . 9 ⊢ (𝑎 · 𝑏) = ( · ‘〈𝑎, 𝑏〉) | |
10 | 9 | eqcomi 2181 | . . . . . . . 8 ⊢ ( · ‘〈𝑎, 𝑏〉) = (𝑎 · 𝑏) |
11 | 10 | eleq1i 2243 | . . . . . . 7 ⊢ (( · ‘〈𝑎, 𝑏〉) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵) |
12 | 8, 11 | bitrdi 196 | . . . . . 6 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵)) |
13 | 12 | ralxp 4770 | . . . . 5 ⊢ (∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
14 | 6, 13 | sylibr 134 | . . . 4 ⊢ (𝑅 ∈ SRing → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
15 | 14 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
16 | fnfvrnss 5676 | . . 3 ⊢ (( · Fn (𝐵 × 𝐵) ∧ ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) → ran · ⊆ 𝐵) | |
17 | 1, 15, 16 | syl2anc 411 | . 2 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ran · ⊆ 𝐵) |
18 | df-f 5220 | . 2 ⊢ ( · :(𝐵 × 𝐵)⟶𝐵 ↔ ( · Fn (𝐵 × 𝐵) ∧ ran · ⊆ 𝐵)) | |
19 | 1, 17, 18 | sylanbrc 417 | 1 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 〈cop 3595 × cxp 4624 ran crn 4627 Fn wfn 5211 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 .rcmulr 12531 SRingcsrg 13099 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
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-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-mgp 13084 df-srg 13100 |
This theorem is referenced by: (None) |
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