![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ricqusker | Structured version Visualization version GIF version |
Description: The image 𝐻 of a ring homomorphism 𝐹 is isomorphic with the quotient ring 𝑄 over 𝐹's kernel 𝐾. This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025.) |
Ref | Expression |
---|---|
rhmqusker.1 | ⊢ 0 = (0g‘𝐻) |
rhmqusker.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) |
rhmqusker.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
rhmqusker.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) |
rhmqusker.s | ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) |
rhmqusker.2 | ⊢ (𝜑 → 𝐺 ∈ CRing) |
Ref | Expression |
---|---|
ricqusker | ⊢ (𝜑 → 𝑄 ≃𝑟 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmqusker.1 | . . 3 ⊢ 0 = (0g‘𝐻) | |
2 | rhmqusker.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) | |
3 | rhmqusker.k | . . 3 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
4 | rhmqusker.q | . . 3 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) | |
5 | rhmqusker.s | . . 3 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) | |
6 | rhmqusker.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CRing) | |
7 | imaeq2 6070 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
8 | 7 | unieqd 4927 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
9 | 8 | cbvmptv 5262 | . . 3 ⊢ (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
10 | 1, 2, 3, 4, 5, 6, 9 | rhmqusker 33397 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 RingIso 𝐻)) |
11 | brrici 20507 | . 2 ⊢ ((𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 RingIso 𝐻) → 𝑄 ≃𝑟 𝐻) | |
12 | 10, 11 | syl 17 | 1 ⊢ (𝜑 → 𝑄 ≃𝑟 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 {csn 4630 ∪ cuni 4914 class class class wbr 5149 ↦ cmpt 5232 ◡ccnv 5682 ran crn 5684 “ cima 5686 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 0gc0g 17475 /s cqus 17541 ~QG cqg 19138 CRingccrg 20237 RingHom crh 20471 RingIso crs 20472 ≃𝑟 cric 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-tpos 8244 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-er 8738 df-ec 8740 df-qs 8744 df-map 8861 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12605 df-dec 12725 df-uz 12870 df-fz 13538 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-0g 17477 df-imas 17544 df-qus 17545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18794 df-submnd 18795 df-grp 18952 df-minusg 18953 df-sbg 18954 df-subg 19139 df-nsg 19140 df-eqg 19141 df-ghm 19229 df-gim 19275 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20156 df-ur 20185 df-ring 20238 df-cring 20239 df-oppr 20336 df-rhm 20474 df-rim 20475 df-ric 20477 df-subrg 20573 df-lmod 20858 df-lss 20929 df-lsp 20969 df-sra 21171 df-rgmod 21172 df-lidl 21217 df-rsp 21218 df-2idl 21259 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |