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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 6084 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝐹 “ 𝑝) = (𝐹 “ 𝑞)) | |
7 | 6 | unieqd 4944 | . . . 4 ⊢ (𝑝 = 𝑞 → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ 𝑞)) |
8 | 7 | cbvmptv 5282 | . . 3 ⊢ (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
9 | 1, 2, 3, 4, 5, 8 | lmhmqusker 33402 | . 2 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑝)) ∈ (𝑄 LMIso 𝐻)) |
10 | brlmici 21086 | . 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 1537 ∈ wcel 2103 {csn 4648 ∪ cuni 4931 class class class wbr 5169 ↦ cmpt 5252 ◡ccnv 5698 ran crn 5700 “ cima 5702 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 0gc0g 17494 /s cqus 17560 ~QG cqg 19157 LMHom clmhm 21036 LMIso clmim 21037 ≃𝑚 clmic 21038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-ec 8761 df-qs 8765 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-0g 17496 df-imas 17563 df-qus 17564 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-nsg 19159 df-eqg 19160 df-ghm 19248 df-gim 19294 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lmim 21040 df-lmic 21041 |
This theorem is referenced by: r1pquslmic 33588 |
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