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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 487 | . 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 19264 | . . . . . . 7 ⊢ ((𝑅 ∈ SRing ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵) |
5 | 4 | 3expb 1116 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵) |
6 | 5 | ralrimivva 3193 | . . . . 5 ⊢ (𝑅 ∈ SRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
7 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → ( · ‘𝑐) = ( · ‘〈𝑎, 𝑏〉)) | |
8 | 7 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ ( · ‘〈𝑎, 𝑏〉) ∈ 𝐵)) |
9 | df-ov 7161 | . . . . . . . . 9 ⊢ (𝑎 · 𝑏) = ( · ‘〈𝑎, 𝑏〉) | |
10 | 9 | eqcomi 2832 | . . . . . . . 8 ⊢ ( · ‘〈𝑎, 𝑏〉) = (𝑎 · 𝑏) |
11 | 10 | eleq1i 2905 | . . . . . . 7 ⊢ (( · ‘〈𝑎, 𝑏〉) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵) |
12 | 8, 11 | syl6bb 289 | . . . . . 6 ⊢ (𝑐 = 〈𝑎, 𝑏〉 → (( · ‘𝑐) ∈ 𝐵 ↔ (𝑎 · 𝑏) ∈ 𝐵)) |
13 | 12 | ralxp 5714 | . . . . 5 ⊢ (∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 · 𝑏) ∈ 𝐵) |
14 | 6, 13 | sylibr 236 | . . . 4 ⊢ (𝑅 ∈ SRing → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) |
16 | fnfvrnss 6886 | . . 3 ⊢ (( · Fn (𝐵 × 𝐵) ∧ ∀𝑐 ∈ (𝐵 × 𝐵)( · ‘𝑐) ∈ 𝐵) → ran · ⊆ 𝐵) | |
17 | 1, 15, 16 | syl2anc 586 | . 2 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → ran · ⊆ 𝐵) |
18 | df-f 6361 | . 2 ⊢ ( · :(𝐵 × 𝐵)⟶𝐵 ↔ ( · Fn (𝐵 × 𝐵) ∧ ran · ⊆ 𝐵)) | |
19 | 1, 17, 18 | sylanbrc 585 | 1 ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 〈cop 4575 × cxp 5555 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 SRingcsrg 19257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mgp 19242 df-srg 19258 |
This theorem is referenced by: (None) |
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