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Mirrors > Home > MPE Home > Th. List > Mathboxes > rimcnv | Structured version Visualization version GIF version |
Description: The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.) |
Ref | Expression |
---|---|
rimcnv | ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | 1, 2 | rhmf 20477 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) |
4 | frel 6739 | . . . . . 6 ⊢ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → Rel 𝐹) | |
5 | dfrel2 6207 | . . . . . 6 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 218 | . . . . 5 ⊢ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → ◡◡𝐹 = 𝐹) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ◡◡𝐹 = 𝐹) |
8 | id 22 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
9 | 7, 8 | eqeltrd 2840 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ◡◡𝐹 ∈ (𝑅 RingHom 𝑆)) |
10 | 9 | anim1ci 616 | . 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → (◡𝐹 ∈ (𝑆 RingHom 𝑅) ∧ ◡◡𝐹 ∈ (𝑅 RingHom 𝑆))) |
11 | isrim0 20475 | . 2 ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅))) | |
12 | isrim0 20475 | . 2 ⊢ (◡𝐹 ∈ (𝑆 RingIso 𝑅) ↔ (◡𝐹 ∈ (𝑆 RingHom 𝑅) ∧ ◡◡𝐹 ∈ (𝑅 RingHom 𝑆))) | |
13 | 10, 11, 12 | 3imtr4i 292 | 1 ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ◡ccnv 5682 Rel wrel 5688 ⟶wf 6555 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 RingHom crh 20461 RingIso crs 20462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-plusg 17306 df-0g 17482 df-mhm 18792 df-ghm 19227 df-mgp 20134 df-ur 20175 df-ring 20228 df-rhm 20464 df-rim 20465 |
This theorem is referenced by: ricsym 42507 |
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