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Mirrors > Home > MPE Home > Th. List > Mathboxes > gicqusker | Structured version Visualization version GIF version |
Description: The image 𝐻 of a group homomorphism 𝐹 is isomorphic with the quotient group 𝑄 over 𝐹's kernel 𝐾. Together with ghmker 19084 and ghmima 19079, this is sometimes called the first isomorphism theorem for groups. (Contributed by Thierry Arnoux, 10-Mar-2025.) |
Ref | Expression |
---|---|
gicqusker.1 | ⊢ 0 = (0g‘𝐻) |
gicqusker.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
gicqusker.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
gicqusker.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) |
gicqusker.s | ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) |
Ref | Expression |
---|---|
gicqusker | ⊢ (𝜑 → 𝑄 ≃𝑔 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gicqusker.1 | . . 3 ⊢ 0 = (0g‘𝐻) | |
2 | gicqusker.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
3 | gicqusker.k | . . 3 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
4 | gicqusker.q | . . 3 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) | |
5 | imaeq2 6045 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
6 | 5 | unieqd 4915 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
7 | 6 | cbvmptv 5254 | . . 3 ⊢ (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
8 | gicqusker.s | . . 3 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) | |
9 | 1, 2, 3, 4, 7, 8 | ghmqusker 32387 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 GrpIso 𝐻)) |
10 | brgici 19110 | . 2 ⊢ ((𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 GrpIso 𝐻) → 𝑄 ≃𝑔 𝐻) | |
11 | 9, 10 | syl 17 | 1 ⊢ (𝜑 → 𝑄 ≃𝑔 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4622 ∪ cuni 4901 class class class wbr 5141 ↦ cmpt 5224 ◡ccnv 5668 ran crn 5670 “ cima 5672 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 0gc0g 17367 /s cqus 17433 ~QG cqg 18974 GrpHom cghm 19055 GrpIso cgim 19097 ≃𝑔 cgic 19098 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-ec 8688 df-qs 8692 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-0g 17369 df-imas 17436 df-qus 17437 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-nsg 18976 df-eqg 18977 df-ghm 19056 df-gim 19099 df-gic 19100 |
This theorem is referenced by: (None) |
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