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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 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ⊆ 𝐵) |
7 | gsummptf1o.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) | |
8 | 6, 7 | sseldd 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
9 | 8 | fmpttd 7130 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
10 | eqid 2728 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | 2 | fvexi 6916 | . . . . 5 ⊢ 0 ∈ V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
13 | 10, 4, 8, 12 | fsuppmptdm 9407 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
14 | gsummptf1o.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) | |
15 | 14 | ralrimiva 3143 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴) |
16 | gsummptf1o.h | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) | |
17 | 16 | ralrimiva 3143 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
18 | eqid 2728 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸) | |
19 | 18 | f1ompt 7126 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴 ↔ (∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)) |
20 | 15, 17, 19 | sylanbrc 581 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴) |
21 | 1, 2, 3, 4, 9, 13, 20 | gsumf1o 19878 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)))) |
22 | eqidd 2729 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸)) | |
23 | eqidd 2729 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
24 | 15, 22, 23 | fmptcos 7146 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶)) |
25 | nfv 1909 | . . . . . 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 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) |
30 | 25, 27, 14, 29 | csbiedf 3925 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ⦋𝐸 / 𝑥⦌𝐶 = 𝐻) |
31 | 30 | mpteq2dva 5252 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
32 | 24, 31 | eqtrd 2768 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
33 | 32 | oveq2d 7442 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸))) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
34 | 21, 33 | eqtrd 2768 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2879 ∀wral 3058 ∃!wreu 3372 Vcvv 3473 ⦋csb 3894 ⊆ wss 3949 ↦ cmpt 5235 ∘ ccom 5686 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 Basecbs 17187 0gc0g 17428 Σg cgsu 17429 CMndccmn 19742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-0g 17430 df-gsum 17431 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-cntz 19275 df-cmn 19744 |
This theorem is referenced by: gsummpt2co 32783 gsumhashmul 32791 mdetpmtr1 33457 |
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