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Mirrors > Home > MPE Home > Th. List > brric2 | Structured version Visualization version GIF version |
Description: The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 35260 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.) |
Ref | Expression |
---|---|
brric2 | ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brric 19498 | . 2 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) | |
2 | n0 4309 | . 2 ⊢ ((𝑅 RingIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)) | |
3 | rimrcl 19475 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V)) | |
4 | isrim0 19474 | . . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑓 ∈ (𝑅 RingIso 𝑆) ↔ (𝑓 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝑓 ∈ (𝑆 RingHom 𝑅)))) | |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
7 | 5, 6 | isrhm 19472 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝑓 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑓 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
8 | 7 | simplbi 500 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 RingHom 𝑆) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
9 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝑓 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝑓 ∈ (𝑆 RingHom 𝑅)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
10 | 4, 9 | syl6bi 255 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑓 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring))) |
11 | 3, 10 | mpcom 38 | . . . 4 ⊢ (𝑓 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
12 | 11 | exlimiv 1927 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring)) |
13 | 12 | pm4.71ri 563 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆))) |
14 | 1, 2, 13 | 3bitri 299 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 class class class wbr 5065 ◡ccnv 5553 ‘cfv 6354 (class class class)co 7155 MndHom cmhm 17953 GrpHom cghm 18354 mulGrpcmgp 19238 Ringcrg 19296 RingHom crh 19463 RingIso crs 19464 ≃𝑟 cric 19465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-mhm 17955 df-ghm 18355 df-mgp 19239 df-ur 19251 df-ring 19298 df-rnghom 19466 df-rngiso 19467 df-ric 19469 |
This theorem is referenced by: ricgic 19500 |
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