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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 483 | . 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 20145 | . . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵) |
5 | 4 | 3expb 1117 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵) |
6 | 5 | ralrimivva 3190 | . . . . 5 ⊢ (𝑅 ∈ SRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
7 | fveq2 6896 | . . . . . . . 8 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → ( · ‘𝑐) = ( · ‘〈𝑎, 𝑏〉)) | |
8 | 7 | eleq1d 2810 | . . . . . . 7 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ ( · ‘〈𝑎, 𝑏〉) ∈ 𝐵)) |
9 | df-ov 7422 | . . . . . . . . 9 ⊢ (𝑎 · 𝑏) = ( · ‘〈𝑎, 𝑏〉) | |
10 | 9 | eqcomi 2734 | . . . . . . . 8 ⊢ ( · ‘〈𝑎, 𝑏〉) = (𝑎 · 𝑏) |
11 | 10 | eleq1i 2816 | . . . . . . 7 ⊢ (( · ‘〈𝑎, 𝑏〉) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵) |
12 | 8, 11 | bitrdi 286 | . . . . . 6 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵)) |
13 | 12 | ralxp 5844 | . . . . 5 ⊢ (∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
14 | 6, 13 | sylibr 233 | . . . 4 ⊢ (𝑅 ∈ SRing → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
15 | 14 | adantr 479 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
16 | fnfvrnss 7130 | . . 3 ⊢ (( · Fn (𝐵 × 𝐵) ∧ ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) → ran · ⊆ 𝐵) | |
17 | 1, 15, 16 | syl2anc 582 | . 2 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ran · ⊆ 𝐵) |
18 | df-f 6553 | . 2 ⊢ ( · :(𝐵 × 𝐵)⟶𝐵 ↔ ( · Fn (𝐵 × 𝐵) ∧ ran · ⊆ 𝐵)) | |
19 | 1, 17, 18 | sylanbrc 581 | 1 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 〈cop 4636 × cxp 5676 ran crn 5679 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 SRingcsrg 20138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mgp 20087 df-srg 20139 |
This theorem is referenced by: (None) |
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