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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 7465 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ V) | |
5 | gsummptshft.a | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝐵) | |
6 | 5 | fmpttd 7134 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴):(𝑀...𝑁)⟶𝐵) |
7 | eqid 2734 | . . . 4 ⊢ (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) | |
8 | fzfid 14010 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
9 | 2 | fvexi 6920 | . . . . 5 ⊢ 0 ∈ V |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
11 | 7, 8, 5, 10 | fsuppmptdm 9413 | . . 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 15810 | . . 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 19948 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))))) |
17 | elfzelz 13560 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℤ) | |
18 | 17 | zcnd 12720 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℂ) |
19 | 12 | zcnd 12720 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
20 | npcan 11514 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) | |
21 | 18, 19, 20 | syl2anr 597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) |
22 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) | |
23 | 21, 22 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
24 | 13, 14 | jca 511 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
25 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
26 | 17 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ℤ) |
27 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝐾 ∈ ℤ) |
28 | 26, 27 | zsubcld 12724 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ ℤ) |
29 | fzaddel 13594 | . . . . . 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 837 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) ∈ (𝑀...𝑁) ↔ ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
31 | 23, 30 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ (𝑀...𝑁)) |
32 | eqidd 2735 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) | |
33 | eqidd 2735 | . . . 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 7148 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶)) |
36 | 35 | oveq2d 7446 | . 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 2774 | 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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ↦ cmpt 5230 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 + caddc 11155 − cmin 11489 ℤcz 12610 ...cfz 13543 Basecbs 17244 0gc0g 17485 Σg cgsu 17486 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-cntz 19347 df-cmn 19814 |
This theorem is referenced by: srgbinomlem4 20246 cpmadugsumlemF 22897 |
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