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Mirrors > Home > MPE Home > Th. List > brric | Structured version Visualization version GIF version |
Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.) |
Ref | Expression |
---|---|
brric | ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ric 20201 | . 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o)) | |
2 | ovex 7423 | . . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
3 | rabexg 5321 | . . . . 5 ⊢ ((𝑟 RingHom 𝑠) ∈ V → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V) | |
4 | 2, 3 | mp1i 13 | . . . 4 ⊢ ((𝑟 ∈ V ∧ 𝑠 ∈ V) → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V) |
5 | 4 | rgen2 3196 | . . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V |
6 | df-rngiso 20199 | . . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)}) | |
7 | 6 | fnmpo 8034 | . . 3 ⊢ (∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V → RingIso Fn (V × V)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ RingIso Fn (V × V) |
9 | 1, 8 | brwitnlem 8486 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 {crab 3429 Vcvv 3470 ∅c0 4315 class class class wbr 5138 × cxp 5664 ◡ccnv 5665 Fn wfn 6524 (class class class)co 7390 RingHom crh 20195 RingIso crs 20196 ≃𝑟 cric 20197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7954 df-2nd 7955 df-1o 8445 df-rngiso 20199 df-ric 20201 |
This theorem is referenced by: brrici 20231 brric2 20232 mat1ric 21913 scmatric 21963 matcpmric 22185 pmmpric 22249 ricsym 40882 rictr 40883 riccrng1 40884 ricdrng1 40892 |
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