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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmicqusker | Structured version Visualization version GIF version |
Description: The image 𝐻 of a module homomorphism 𝐹 is isomorphic with the quotient module 𝑄 over 𝐹's kernel 𝐾. This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025.) |
Ref | Expression |
---|---|
lmhmqusker.1 | ⊢ 0 = (0g‘𝐻) |
lmhmqusker.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻)) |
lmhmqusker.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
lmhmqusker.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) |
lmhmqusker.s | ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) |
Ref | Expression |
---|---|
lmicqusker | ⊢ (𝜑 → 𝑄 ≃𝑚 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmqusker.1 | . . 3 ⊢ 0 = (0g‘𝐻) | |
2 | lmhmqusker.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻)) | |
3 | lmhmqusker.k | . . 3 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
4 | lmhmqusker.q | . . 3 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) | |
5 | lmhmqusker.s | . . 3 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) | |
6 | imaeq2 6045 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
7 | 6 | unieqd 4912 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
8 | 7 | cbvmptv 5251 | . . 3 ⊢ (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
9 | 1, 2, 3, 4, 5, 8 | lmhmqusker 33003 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 LMIso 𝐻)) |
10 | brlmici 20907 | . 2 ⊢ ((𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 LMIso 𝐻) → 𝑄 ≃𝑚 𝐻) | |
11 | 9, 10 | syl 17 | 1 ⊢ (𝜑 → 𝑄 ≃𝑚 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4620 ∪ cuni 4899 class class class wbr 5138 ↦ cmpt 5221 ◡ccnv 5665 ran crn 5667 “ cima 5669 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 0gc0g 17384 /s cqus 17450 ~QG cqg 19039 LMHom clmhm 20857 LMIso clmim 20858 ≃𝑚 clmic 20859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 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 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-0g 17386 df-imas 17453 df-qus 17454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-gim 19174 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-lmod 20698 df-lss 20769 df-lmhm 20860 df-lmim 20861 df-lmic 20862 |
This theorem is referenced by: r1pquslmic 33147 |
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