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Mirrors > Home > MPE Home > Th. List > gsummptshft | Structured version Visualization version GIF version |
Description: Index shift of a finite group sum over a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
Ref | Expression |
---|---|
gsummptshft.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptshft.z | ⊢ 0 = (0g‘𝐺) |
gsummptshft.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptshft.k | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
gsummptshft.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
gsummptshft.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
gsummptshft.a | ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝐵) |
gsummptshft.c | ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
gsummptshft | ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptshft.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptshft.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsummptshft.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | ovexd 7183 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ V) | |
5 | gsummptshft.a | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝐵) | |
6 | 5 | fmpttd 6872 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴):(𝑀...𝑁)⟶𝐵) |
7 | eqid 2819 | . . . 4 ⊢ (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) | |
8 | fzfid 13333 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
9 | 2 | fvexi 6677 | . . . . 5 ⊢ 0 ∈ V |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
11 | 7, 8, 5, 10 | fsuppmptdm 8836 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) finSupp 0 ) |
12 | gsummptshft.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
13 | gsummptshft.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
14 | gsummptshft.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | 12, 13, 14 | mptfzshft 15125 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁)) |
16 | 1, 2, 3, 4, 6, 11, 15 | gsumf1o 19028 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))))) |
17 | elfzelz 12900 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℤ) | |
18 | 17 | zcnd 12080 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℂ) |
19 | 12 | zcnd 12080 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
20 | npcan 10887 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) | |
21 | 18, 19, 20 | syl2anr 598 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) |
22 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) | |
23 | 21, 22 | eqeltrd 2911 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
24 | 13, 14 | jca 514 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
25 | 24 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
26 | 17 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ℤ) |
27 | 12 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝐾 ∈ ℤ) |
28 | 26, 27 | zsubcld 12084 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ ℤ) |
29 | fzaddel 12933 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑘 − 𝐾) ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑘 − 𝐾) ∈ (𝑀...𝑁) ↔ ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) | |
30 | 25, 28, 27, 29 | syl12anc 834 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) ∈ (𝑀...𝑁) ↔ ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
31 | 23, 30 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ (𝑀...𝑁)) |
32 | eqidd 2820 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) | |
33 | eqidd 2820 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) | |
34 | gsummptshft.c | . . . 4 ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐶) | |
35 | 31, 32, 33, 34 | fmptco 6884 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶)) |
36 | 35 | oveq2d 7164 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)))) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶))) |
37 | 16, 36 | eqtrd 2854 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ↦ cmpt 5137 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 + caddc 10532 − cmin 10862 ℤcz 11973 ...cfz 12884 Basecbs 16475 0gc0g 16705 Σg cgsu 16706 CMndccmn 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: srgbinomlem4 19285 cpmadugsumlemF 21476 |
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