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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 | ovex 6577 | . . . 4 ⊢ (𝑀...𝑁) ∈ V | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ V) |
6 | gsummptshft.a | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝐵) | |
7 | eqid 2610 | . . . 4 ⊢ (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) | |
8 | 6, 7 | fmptd 6292 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴):(𝑀...𝑁)⟶𝐵) |
9 | fzfid 12634 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
10 | fvex 6113 | . . . . . 6 ⊢ (0g‘𝐺) ∈ V | |
11 | 2, 10 | eqeltri 2684 | . . . . 5 ⊢ 0 ∈ V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
13 | 7, 9, 6, 12 | fsuppmptdm 8169 | . . 3 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) finSupp 0 ) |
14 | gsummptshft.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
15 | gsummptshft.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
16 | gsummptshft.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
17 | 14, 15, 16 | mptfzshft 14352 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁)) |
18 | 1, 2, 3, 5, 8, 13, 17 | gsumf1o 18140 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))))) |
19 | elfzelz 12213 | . . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℤ) | |
20 | 19 | zcnd 11359 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑘 ∈ ℂ) |
21 | 14 | zcnd 11359 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
22 | npcan 10169 | . . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) | |
23 | 20, 21, 22 | syl2anr 494 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) = 𝑘) |
24 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) | |
25 | 23, 24 | eqeltrd 2688 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
26 | 15, 16 | jca 553 | . . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
27 | 26 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
28 | 19 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑘 ∈ ℤ) |
29 | 14 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝐾 ∈ ℤ) |
30 | 28, 29 | zsubcld 11363 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ ℤ) |
31 | fzaddel 12246 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑘 − 𝐾) ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑘 − 𝐾) ∈ (𝑀...𝑁) ↔ ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) | |
32 | 27, 30, 29, 31 | syl12anc 1316 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → ((𝑘 − 𝐾) ∈ (𝑀...𝑁) ↔ ((𝑘 − 𝐾) + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
33 | 25, 32 | mpbird 246 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑘 − 𝐾) ∈ (𝑀...𝑁)) |
34 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) | |
35 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) | |
36 | gsummptshft.c | . . . 4 ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐶) | |
37 | 33, 34, 35, 36 | fmptco 6303 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾))) = (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶)) |
38 | 37 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑗 ∈ (𝑀...𝑁) ↦ 𝐴) ∘ (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑘 − 𝐾)))) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶))) |
39 | 18, 38 | eqtrd 2644 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 + caddc 9818 − cmin 10145 ℤcz 11254 ...cfz 12197 Basecbs 15695 0gc0g 15923 Σg cgsu 15924 CMndccmn 18016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cntz 17573 df-cmn 18018 |
This theorem is referenced by: srgbinomlem4 18366 cpmadugsumlemF 20500 |
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