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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 20492 | . 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o)) | |
2 | ovex 7464 | . . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
3 | rabexg 5343 | . . . . 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 3197 | . . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V |
6 | df-rim 20490 | . . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)}) | |
7 | 6 | fnmpo 8093 | . . 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 8544 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 Vcvv 3478 ∅c0 4339 class class class wbr 5148 × cxp 5687 ◡ccnv 5688 Fn wfn 6558 (class class class)co 7431 RingHom crh 20486 RingIso crs 20487 ≃𝑟 cric 20488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-rim 20490 df-ric 20492 |
This theorem is referenced by: brrici 20522 brric2 20523 mat1ric 22509 scmatric 22559 matcpmric 22781 pmmpric 22845 ricsym 42506 rictr 42507 riccrng1 42508 ricdrng1 42515 |
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