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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 6984 | . . 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 19535 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
11 | eqidd 2741 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
12 | eqidd 2741 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
13 | gsummhm2.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) | |
14 | 7, 11, 12, 13 | fmptco 6996 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝑘 ∈ 𝐴 ↦ 𝐷)) |
15 | 14 | oveq2d 7285 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷))) |
16 | eqid 2740 | . . 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 19512 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) ∈ 𝐵) |
19 | 17 | eleq1d 2825 | . . . 4 ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝐶 ∈ (Base‘𝐻) ↔ 𝐸 ∈ (Base‘𝐻))) |
20 | eqid 2740 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
21 | 1, 20 | mhmf 18431 | . . . . . 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 6979 | . . . . 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 3563 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐻)) |
26 | 16, 17, 18, 25 | fvmptd3 6893 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) = 𝐸) |
27 | 10, 15, 26 | 3eqtr3d 2788 | 1 ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 class class class wbr 5079 ↦ cmpt 5162 ∘ ccom 5593 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 finSupp cfsupp 9104 Basecbs 16908 0gc0g 17146 Σg cgsu 17147 Mndcmnd 18381 MndHom cmhm 18424 CMndccmn 19382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-seq 13718 df-hash 14041 df-0g 17148 df-gsum 17149 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mhm 18426 df-cntz 18919 df-cmn 19384 |
This theorem is referenced by: gsummulglem 19538 prdsgsum 19578 srgsummulcr 19769 sgsummulcl 19770 gsummulc1 19841 gsummulc2 19842 gsumvsmul 20183 lgseisenlem4 26522 gsumvsmul1 31305 pwsgprod 40264 mhphflem 40279 |
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