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Mirrors > Home > MPE Home > Th. List > ringciso | Structured version Visualization version GIF version |
Description: An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.) |
Ref | Expression |
---|---|
ringcsect.c | ⊢ 𝐶 = (RingCat‘𝑈) |
ringcsect.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcsect.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringcsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringcsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringciso.n | ⊢ 𝐼 = (Iso‘𝐶) |
Ref | Expression |
---|---|
ringciso | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcsect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2725 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | ringcsect.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | ringcsect.c | . . . . . 6 ⊢ 𝐶 = (RingCat‘𝑈) | |
5 | 4 | ringccat 20608 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | ringcsect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | ringcsect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | ringciso.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
10 | 1, 2, 6, 7, 8, 9 | isoval 17751 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
11 | 10 | eleq2d 2811 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
12 | 1, 2, 6, 7, 8 | invfun 17750 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌)) |
13 | funfvbrb 7059 | . . . . 5 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) |
15 | 4, 1, 3, 7, 8, 2 | ringcinv 20616 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹))) |
16 | simpl 481 | . . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌)) | |
17 | 15, 16 | biimtrdi 252 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌))) |
18 | 14, 17 | sylbid 239 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RingIso 𝑌))) |
19 | eqid 2725 | . . . 4 ⊢ ◡𝐹 = ◡𝐹 | |
20 | 4, 1, 3, 7, 8, 2 | ringcinv 20616 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ◡𝐹 = ◡𝐹))) |
21 | funrel 6571 | . . . . . . 7 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → Rel (𝑋(Inv‘𝐶)𝑌)) | |
22 | 12, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → Rel (𝑋(Inv‘𝐶)𝑌)) |
23 | releldm 5946 | . . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)) | |
24 | 23 | ex 411 | . . . . . 6 ⊢ (Rel (𝑋(Inv‘𝐶)𝑌) → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
25 | 22, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
26 | 20, 25 | sylbird 259 | . . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ◡𝐹 = ◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
27 | 19, 26 | mpan2i 695 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
28 | 18, 27 | impbid 211 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
29 | 11, 28 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ◡ccnv 5677 dom cdm 5678 Rel wrel 5683 Fun wfun 6543 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Catccat 17647 Invcinv 17731 Isociso 17732 RingIso crs 20421 RingCatcringc 20590 |
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-rep 5286 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-rmo 3363 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-tp 4635 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-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 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-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-hom 17260 df-cco 17261 df-0g 17426 df-cat 17651 df-cid 17652 df-homf 17653 df-sect 17733 df-inv 17734 df-iso 17735 df-ssc 17796 df-resc 17797 df-subc 17798 df-estrc 18116 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-grp 18901 df-ghm 19176 df-mgp 20087 df-ur 20134 df-ring 20187 df-rhm 20423 df-rim 20424 df-ringc 20591 |
This theorem is referenced by: (None) |
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