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Mirrors > Home > MPE Home > Th. List > gsummptf1o | Structured version Visualization version GIF version |
Description: Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.) |
Ref | Expression |
---|---|
gsummptf1o.x | ⊢ Ⅎ𝑥𝐻 |
gsummptf1o.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptf1o.z | ⊢ 0 = (0g‘𝐺) |
gsummptf1o.i | ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) |
gsummptf1o.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptf1o.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptf1o.d | ⊢ (𝜑 → 𝐹 ⊆ 𝐵) |
gsummptf1o.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) |
gsummptf1o.e | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) |
gsummptf1o.h | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
Ref | Expression |
---|---|
gsummptf1o | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptf1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptf1o.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsummptf1o.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummptf1o.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | gsummptf1o.d | . . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝐵) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ⊆ 𝐵) |
7 | gsummptf1o.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) | |
8 | 6, 7 | sseldd 3916 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
9 | 8 | fmpttd 6856 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
10 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | 2 | fvexi 6659 | . . . . 5 ⊢ 0 ∈ V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
13 | 10, 4, 8, 12 | fsuppmptdm 8828 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
14 | gsummptf1o.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) | |
15 | 14 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴) |
16 | gsummptf1o.h | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) | |
17 | 16 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
18 | eqid 2798 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸) | |
19 | 18 | f1ompt 6852 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴 ↔ (∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)) |
20 | 15, 17, 19 | sylanbrc 586 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴) |
21 | 1, 2, 3, 4, 9, 13, 20 | gsumf1o 19029 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)))) |
22 | eqidd 2799 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸)) | |
23 | eqidd 2799 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
24 | 15, 22, 23 | fmptcos 6870 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶)) |
25 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐷) | |
26 | gsummptf1o.x | . . . . . . 7 ⊢ Ⅎ𝑥𝐻 | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → Ⅎ𝑥𝐻) |
28 | gsummptf1o.i | . . . . . . 7 ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) | |
29 | 28 | adantl 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) |
30 | 25, 27, 14, 29 | csbiedf 3858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ⦋𝐸 / 𝑥⦌𝐶 = 𝐻) |
31 | 30 | mpteq2dva 5125 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
32 | 24, 31 | eqtrd 2833 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
33 | 32 | oveq2d 7151 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸))) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
34 | 21, 33 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃!wreu 3108 Vcvv 3441 ⦋csb 3828 ⊆ wss 3881 ↦ cmpt 5110 ∘ ccom 5523 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 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: gsummpt2co 30733 gsumhashmul 30741 mdetpmtr1 31176 |
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