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Mirrors > Home > MPE Home > Th. List > issrng | Structured version Visualization version GIF version |
Description: The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
issrng.o | ⊢ 𝑂 = (oppr‘𝑅) |
issrng.i | ⊢ ∗ = (*rf‘𝑅) |
Ref | Expression |
---|---|
issrng | ⊢ (𝑅 ∈ *-Ring ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-srng 18846 | . . 3 ⊢ *-Ring = {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)} | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝑅 ∈ *-Ring ↔ 𝑅 ∈ {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)}) |
3 | rhmrcl1 18719 | . . . . 5 ⊢ ( ∗ ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ Ring) | |
4 | elex 3212 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ ( ∗ ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ V) |
6 | 5 | adantr 481 | . . 3 ⊢ (( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ) → 𝑅 ∈ V) |
7 | fvexd 6203 | . . . 4 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) ∈ V) | |
8 | id 22 | . . . . . . 7 ⊢ (𝑖 = (*rf‘𝑟) → 𝑖 = (*rf‘𝑟)) | |
9 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) = (*rf‘𝑅)) | |
10 | issrng.i | . . . . . . . 8 ⊢ ∗ = (*rf‘𝑅) | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) = ∗ ) |
12 | 8, 11 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → 𝑖 = ∗ ) |
13 | simpl 473 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → 𝑟 = 𝑅) | |
14 | 13 | fveq2d 6195 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (oppr‘𝑟) = (oppr‘𝑅)) |
15 | issrng.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
16 | 14, 15 | syl6eqr 2674 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (oppr‘𝑟) = 𝑂) |
17 | 13, 16 | oveq12d 6668 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑟 RingHom (oppr‘𝑟)) = (𝑅 RingHom 𝑂)) |
18 | 12, 17 | eleq12d 2695 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ↔ ∗ ∈ (𝑅 RingHom 𝑂))) |
19 | 12 | cnveqd 5298 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → ◡𝑖 = ◡ ∗ ) |
20 | 12, 19 | eqeq12d 2637 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑖 = ◡𝑖 ↔ ∗ = ◡ ∗ )) |
21 | 18, 20 | anbi12d 747 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → ((𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖) ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ))) |
22 | 7, 21 | sbcied 3472 | . . 3 ⊢ (𝑟 = 𝑅 → ([(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖) ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ))) |
23 | 6, 22 | elab3 3358 | . 2 ⊢ (𝑅 ∈ {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)} ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) |
24 | 2, 23 | bitri 264 | 1 ⊢ (𝑅 ∈ *-Ring ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 [wsbc 3435 ◡ccnv 5113 ‘cfv 5888 (class class class)co 6650 Ringcrg 18547 opprcoppr 18622 RingHom crh 18712 *rfcstf 18843 *-Ringcsr 18844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mhm 17335 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-rnghom 18715 df-srng 18846 |
This theorem is referenced by: srngrhm 18851 srngcnv 18853 issrngd 18861 |
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