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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 6881 | . . 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 19060 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) |
11 | eqidd 2824 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
12 | eqidd 2824 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
13 | gsummhm2.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) | |
14 | 7, 11, 12, 13 | fmptco 6893 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝑘 ∈ 𝐴 ↦ 𝐷)) |
15 | 14 | oveq2d 7174 | . 2 ⊢ (𝜑 → (𝐻 Σg ((𝑥 ∈ 𝐵 ↦ 𝐶) ∘ (𝑘 ∈ 𝐴 ↦ 𝑋))) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷))) |
16 | eqid 2823 | . . 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 19037 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) ∈ 𝐵) |
19 | 17 | eleq1d 2899 | . . . 4 ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝐶 ∈ (Base‘𝐻) ↔ 𝐸 ∈ (Base‘𝐻))) |
20 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
21 | 1, 20 | mhmf 17963 | . . . . . 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 6876 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝐶 ∈ (Base‘𝐻) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):𝐵⟶(Base‘𝐻)) |
24 | 22, 23 | sylibr 236 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ (Base‘𝐻)) |
25 | 19, 24, 18 | rspcdva 3627 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐻)) |
26 | 16, 17, 18, 25 | fvmptd3 6793 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘(𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) = 𝐸) |
27 | 10, 15, 26 | 3eqtr3d 2866 | 1 ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 ↦ cmpt 5148 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 finSupp cfsupp 8835 Basecbs 16485 0gc0g 16715 Σg cgsu 16716 Mndcmnd 17913 MndHom cmhm 17956 CMndccmn 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-cntz 18449 df-cmn 18910 |
This theorem is referenced by: gsummulglem 19063 prdsgsum 19103 srgsummulcr 19289 sgsummulcl 19290 gsummulc1 19358 gsummulc2 19359 gsumvsmul 19700 lgseisenlem4 25956 gsumvsmul1 30691 |
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