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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngciso | Structured version Visualization version GIF version |
Description: An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) |
Ref | Expression |
---|---|
rngcsect.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcsect.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcsect.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngcsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rngciso.n | ⊢ 𝐼 = (Iso‘𝐶) |
Ref | Expression |
---|---|
rngciso | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcsect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2738 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | rngcsect.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | rngcsect.c | . . . . . 6 ⊢ 𝐶 = (RngCat‘𝑈) | |
5 | 4 | rngccat 45070 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | rngcsect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | rngcsect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | rngciso.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
10 | 1, 2, 6, 7, 8, 9 | isoval 17140 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
11 | 10 | eleq2d 2818 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
12 | 1, 2, 6, 7, 8 | invfun 17139 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌)) |
13 | funfvbrb 6828 | . . . . 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 | rngcinv 45073 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹))) |
16 | simpl 486 | . . . . 5 ⊢ ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹) → 𝐹 ∈ (𝑋 RngIsom 𝑌)) | |
17 | 15, 16 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RngIsom 𝑌))) |
18 | 14, 17 | sylbid 243 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RngIsom 𝑌))) |
19 | eqid 2738 | . . . 4 ⊢ ◡𝐹 = ◡𝐹 | |
20 | 4, 1, 3, 7, 8, 2 | rngcinv 45073 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ ◡𝐹 = ◡𝐹))) |
21 | funrel 6356 | . . . . . . 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 5787 | . . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)) | |
24 | 23 | ex 416 | . . . . . 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 263 | . . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ ◡𝐹 = ◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
27 | 19, 26 | mpan2i 697 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RngIsom 𝑌) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
28 | 18, 27 | impbid 215 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌))) |
29 | 11, 28 | bitrd 282 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ◡ccnv 5524 dom cdm 5525 Rel wrel 5530 Fun wfun 6333 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Catccat 17038 Invcinv 17120 Isociso 17121 RngIsom crngs 44978 RngCatcrngc 45049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-hom 16692 df-cco 16693 df-0g 16818 df-cat 17042 df-cid 17043 df-homf 17044 df-sect 17122 df-inv 17123 df-iso 17124 df-ssc 17185 df-resc 17186 df-subc 17187 df-estrc 17489 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-grp 18222 df-ghm 18474 df-abl 19027 df-mgp 19359 df-mgmhm 44867 df-rng0 44967 df-rnghomo 44979 df-rngisom 44980 df-rngc 45051 |
This theorem is referenced by: (None) |
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