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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 6989 | . . 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 19539 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
11 | eqidd 2739 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
12 | eqidd 2739 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
13 | gsummhm2.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) | |
14 | 7, 11, 12, 13 | fmptco 7001 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝑘 ∈ 𝐴 ↦ 𝐷)) |
15 | 14 | oveq2d 7291 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷))) |
16 | eqid 2738 | . . 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 19516 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) ∈ 𝐵) |
19 | 17 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝐶 ∈ (Base‘𝐻) ↔ 𝐸 ∈ (Base‘𝐻))) |
20 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
21 | 1, 20 | mhmf 18435 | . . . . . 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 6984 | . . . . 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 3562 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐻)) |
26 | 16, 17, 18, 25 | fvmptd3 6898 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) = 𝐸) |
27 | 10, 15, 26 | 3eqtr3d 2786 | 1 ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 ↦ cmpt 5157 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 finSupp cfsupp 9128 Basecbs 16912 0gc0g 17150 Σg cgsu 17151 Mndcmnd 18385 MndHom cmhm 18428 CMndccmn 19386 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 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 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-cntz 18923 df-cmn 19388 |
This theorem is referenced by: gsummulglem 19542 prdsgsum 19582 srgsummulcr 19773 sgsummulcl 19774 gsummulc1 19845 gsummulc2 19846 gsumvsmul 20187 lgseisenlem4 26526 gsumvsmul1 31311 pwsgprod 40269 mhphflem 40284 |
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