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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 19472 | . 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o)) | |
2 | ovex 7191 | . . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
3 | rabexg 5236 | . . . . 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 3205 | . . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V |
6 | df-rngiso 19470 | . . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)}) | |
7 | 6 | fnmpo 7769 | . . 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 8134 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 Vcvv 3496 ∅c0 4293 class class class wbr 5068 × cxp 5555 ◡ccnv 5556 Fn wfn 6352 (class class class)co 7158 RingHom crh 19466 RingIso crs 19467 ≃𝑟 cric 19468 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-rngiso 19470 df-ric 19472 |
This theorem is referenced by: brric2 19502 mat1ric 21098 scmatric 21148 matcpmric 21369 pmmpric 21433 |
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