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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmqusker | Structured version Visualization version GIF version |
Description: A surjective ring homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-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) |
rhmqusker.j | ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
Ref | Expression |
---|---|
rhmqusker | ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingIso 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.j | . . 3 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) | |
6 | rhmqusker.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CRing) | |
7 | 1, 2, 3, 4, 5, 6 | rhmquskerlem 33440 | . 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻)) |
8 | rhmghm 20476 | . . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
10 | rhmqusker.s | . . . 4 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) | |
11 | 1, 9, 3, 4, 5, 10 | ghmqusker 19301 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻)) |
12 | eqid 2736 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
13 | eqid 2736 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
14 | 12, 13 | gimf1o 19277 | . . 3 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)) |
15 | 11, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)) |
16 | 12, 13 | isrim 20484 | . 2 ⊢ (𝐽 ∈ (𝑄 RingIso 𝐻) ↔ (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))) |
17 | 7, 15, 16 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingIso 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4624 ∪ cuni 4905 ↦ cmpt 5223 ◡ccnv 5682 ran crn 5684 “ cima 5686 –1-1-onto→wf1o 6558 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 0gc0g 17480 /s cqus 17546 ~QG cqg 19136 GrpHom cghm 19226 GrpIso cgim 19271 CRingccrg 20227 RingHom crh 20461 RingIso crs 20462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 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 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-ec 8743 df-qs 8747 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9478 df-inf 9479 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-0g 17482 df-imas 17549 df-qus 17550 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-mhm 18792 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-nsg 19138 df-eqg 19139 df-ghm 19227 df-gim 19273 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-cring 20229 df-oppr 20326 df-rhm 20464 df-rim 20465 df-subrg 20562 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21210 df-rsp 21211 df-2idl 21252 |
This theorem is referenced by: ricqusker 33442 |
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