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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 19546 | . . 3 ⊢ *-Ring = {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝑅 ∈ *-Ring ↔ 𝑅 ∈ {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)}) |
3 | rhmrcl1 19400 | . . . . 5 ⊢ ( ∗ ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ Ring) | |
4 | 3 | elexd 3512 | . . . 4 ⊢ ( ∗ ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ V) |
5 | 4 | adantr 481 | . . 3 ⊢ (( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ) → 𝑅 ∈ V) |
6 | fvexd 6678 | . . . 4 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) ∈ V) | |
7 | id 22 | . . . . . . 7 ⊢ (𝑖 = (*rf‘𝑟) → 𝑖 = (*rf‘𝑟)) | |
8 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) = (*rf‘𝑅)) | |
9 | issrng.i | . . . . . . . 8 ⊢ ∗ = (*rf‘𝑅) | |
10 | 8, 9 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (*rf‘𝑟) = ∗ ) |
11 | 7, 10 | sylan9eqr 2875 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → 𝑖 = ∗ ) |
12 | simpl 483 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → 𝑟 = 𝑅) | |
13 | 12 | fveq2d 6667 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (oppr‘𝑟) = (oppr‘𝑅)) |
14 | issrng.o | . . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅) | |
15 | 13, 14 | syl6eqr 2871 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (oppr‘𝑟) = 𝑂) |
16 | 12, 15 | oveq12d 7163 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑟 RingHom (oppr‘𝑟)) = (𝑅 RingHom 𝑂)) |
17 | 11, 16 | eleq12d 2904 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ↔ ∗ ∈ (𝑅 RingHom 𝑂))) |
18 | 11 | cnveqd 5739 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → ◡𝑖 = ◡ ∗ ) |
19 | 11, 18 | eqeq12d 2834 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → (𝑖 = ◡𝑖 ↔ ∗ = ◡ ∗ )) |
20 | 17, 19 | anbi12d 630 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = (*rf‘𝑟)) → ((𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖) ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ))) |
21 | 6, 20 | sbcied 3811 | . . 3 ⊢ (𝑟 = 𝑅 → ([(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖) ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ))) |
22 | 5, 21 | elab3 3671 | . 2 ⊢ (𝑅 ∈ {𝑟 ∣ [(*rf‘𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr‘𝑟)) ∧ 𝑖 = ◡𝑖)} ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) |
23 | 2, 22 | bitri 276 | 1 ⊢ (𝑅 ∈ *-Ring ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 [wsbc 3769 ◡ccnv 5547 ‘cfv 6348 (class class class)co 7145 Ringcrg 19226 opprcoppr 19301 RingHom crh 19393 *rfcstf 19543 *-Ringcsr 19544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mhm 17944 df-ghm 18294 df-mgp 19169 df-ur 19181 df-ring 19228 df-rnghom 19396 df-srng 19546 |
This theorem is referenced by: srngrhm 19551 srngcnv 19553 issrngd 19561 |
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