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Mathbox for Thierry Arnoux |
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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 19103 and ghmima 19098, 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 6048 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
6 | 5 | unieqd 4918 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
7 | 6 | cbvmptv 5257 | . . 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 32487 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 GrpIso 𝐻)) |
10 | brgici 19129 | . 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 1542 ∈ wcel 2107 {csn 4624 ∪ cuni 4904 class class class wbr 5144 ↦ cmpt 5227 ◡ccnv 5671 ran crn 5673 “ cima 5675 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 0gc0g 17372 /s cqus 17438 ~QG cqg 18987 GrpHom cghm 19074 GrpIso cgim 19116 ≃𝑔 cgic 19117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-ec 8693 df-qs 8697 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-0g 17374 df-imas 17441 df-qus 17442 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-nsg 18989 df-eqg 18990 df-ghm 19075 df-gim 19118 df-gic 19119 |
This theorem is referenced by: (None) |
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