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Mirrors > Home > MPE Home > Th. List > gsummhm2 | Structured version Visualization version GIF version |
Description: Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsummhm2.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummhm2.z | ⊢ 0 = (0g‘𝐺) |
gsummhm2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummhm2.h | ⊢ (𝜑 → 𝐻 ∈ Mnd) |
gsummhm2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummhm2.k | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻)) |
gsummhm2.f | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsummhm2.w | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) |
gsummhm2.1 | ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) |
gsummhm2.2 | ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
gsummhm2 | ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummhm2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummhm2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsummhm2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummhm2.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) | |
5 | gsummhm2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | gsummhm2.k | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻)) | |
7 | gsummhm2.f | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
8 | 7 | fmpttd 7067 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵) |
9 | gsummhm2.w | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
10 | 1, 2, 3, 4, 5, 6, 8, 9 | gsummhm 19723 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
11 | eqidd 2734 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
12 | eqidd 2734 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
13 | gsummhm2.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) | |
14 | 7, 11, 12, 13 | fmptco 7079 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝑘 ∈ 𝐴 ↦ 𝐷)) |
15 | 14 | oveq2d 7377 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷))) |
16 | eqid 2733 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
17 | gsummhm2.2 | . . 3 ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → 𝐶 = 𝐸) | |
18 | 1, 2, 3, 5, 8, 9 | gsumcl 19700 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) ∈ 𝐵) |
19 | 17 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝐶 ∈ (Base‘𝐻) ↔ 𝐸 ∈ (Base‘𝐻))) |
20 | eqid 2733 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
21 | 1, 20 | mhmf 18615 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻) → (𝑥 ∈ 𝐵 ↦ 𝐶):𝐵⟶(Base‘𝐻)) |
22 | 6, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶):𝐵⟶(Base‘𝐻)) |
23 | 16 | fmpt 7062 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝐶 ∈ (Base‘𝐻) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):𝐵⟶(Base‘𝐻)) |
24 | 22, 23 | sylibr 233 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ (Base‘𝐻)) |
25 | 19, 24, 18 | rspcdva 3584 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐻)) |
26 | 16, 17, 18, 25 | fvmptd3 6975 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) = 𝐸) |
27 | 10, 15, 26 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 class class class wbr 5109 ↦ cmpt 5192 ∘ ccom 5641 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 finSupp cfsupp 9311 Basecbs 17091 0gc0g 17329 Σg cgsu 17330 Mndcmnd 18564 MndHom cmhm 18607 CMndccmn 19570 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-0g 17331 df-gsum 17332 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-cntz 19105 df-cmn 19572 |
This theorem is referenced by: gsummulglem 19726 prdsgsum 19766 srgsummulcr 19962 sgsummulcl 19963 gsummulc1 20038 gsummulc2 20039 gsumvsmul 20430 lgseisenlem4 26749 gsumvsmul1 31949 pwsgprod 40779 mhphflem 40817 |
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