![]() |
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 6046 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
8 | 7 | unieqd 4916 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
9 | 8 | cbvmptv 5255 | . . 3 ⊢ (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
10 | 1, 2, 3, 4, 5, 6, 9 | rhmqusker 32459 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 RingIso 𝐻)) |
11 | brrici 20236 | . 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 1541 ∈ wcel 2106 {csn 4623 ∪ cuni 4902 class class class wbr 5142 ↦ cmpt 5225 ◡ccnv 5669 ran crn 5671 “ cima 5673 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 0gc0g 17369 /s cqus 17435 ~QG cqg 18976 CRingccrg 20017 RingHom crh 20200 RingIso crs 20201 ≃𝑟 cric 20202 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-ec 8690 df-qs 8694 df-map 8807 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-sup 9421 df-inf 9422 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-fz 13469 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-0g 17371 df-imas 17438 df-qus 17439 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mhm 18649 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-subg 18977 df-nsg 18978 df-eqg 18979 df-ghm 19058 df-gim 19101 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-ring 20018 df-cring 20019 df-oppr 20104 df-rnghom 20203 df-rngiso 20204 df-ric 20206 df-subrg 20312 df-lmod 20424 df-lss 20494 df-lsp 20534 df-sra 20736 df-rgmod 20737 df-lidl 20738 df-rsp 20739 df-2idl 20805 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |