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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringcisoALTV | Structured version Visualization version GIF version |
Description: An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringcsectALTV.c | ⊢ 𝐶 = (RingCatALTV‘𝑈) |
ringcsectALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcsectALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringcsectALTV.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringcsectALTV.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringcisoALTV.n | ⊢ 𝐼 = (Iso‘𝐶) |
Ref | Expression |
---|---|
ringcisoALTV | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcsectALTV.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2821 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | ringcsectALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | ringcsectALTV.c | . . . . . 6 ⊢ 𝐶 = (RingCatALTV‘𝑈) | |
5 | 4 | ringccatALTV 44344 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | ringcsectALTV.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | ringcsectALTV.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | ringcisoALTV.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
10 | 1, 2, 6, 7, 8, 9 | isoval 17035 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
11 | 10 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
12 | 1, 2, 6, 7, 8 | invfun 17034 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌)) |
13 | funfvbrb 6821 | . . . . 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 | ringcinvALTV 44347 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹))) |
16 | simpl 485 | . . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌)) | |
17 | 15, 16 | syl6bi 255 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌))) |
18 | 14, 17 | sylbid 242 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RingIso 𝑌))) |
19 | eqid 2821 | . . . 4 ⊢ ◡𝐹 = ◡𝐹 | |
20 | 4, 1, 3, 7, 8, 2 | ringcinvALTV 44347 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ◡𝐹 = ◡𝐹))) |
21 | funrel 6372 | . . . . . . 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 5814 | . . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)) | |
24 | 23 | ex 415 | . . . . . 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 262 | . . . 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 214 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
29 | 11, 28 | bitrd 281 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ◡ccnv 5554 dom cdm 5555 Rel wrel 5560 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Catccat 16935 Invcinv 17015 Isociso 17016 RingIso crs 19465 RingCatALTVcringcALTV 44295 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-hom 16589 df-cco 16590 df-0g 16715 df-cat 16939 df-cid 16940 df-sect 17017 df-inv 17018 df-iso 17019 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-rnghom 19467 df-rngiso 19468 df-ringcALTV 44297 |
This theorem is referenced by: (None) |
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