![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ⊆ 𝐵) |
7 | gsummptf1o.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) | |
8 | 6, 7 | sseldd 3829 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
9 | 8 | fmpttd 6635 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
10 | eqid 2826 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | 2 | fvexi 6448 | . . . . 5 ⊢ 0 ∈ V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
13 | 10, 4, 8, 12 | fsuppmptdm 8556 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
14 | gsummptf1o.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) | |
15 | 14 | ralrimiva 3176 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴) |
16 | gsummptf1o.h | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) | |
17 | 16 | ralrimiva 3176 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
18 | eqid 2826 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸) | |
19 | 18 | f1ompt 6631 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴 ↔ (∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)) |
20 | 15, 17, 19 | sylanbrc 580 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴) |
21 | 1, 2, 3, 4, 9, 13, 20 | gsumf1o 18671 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)))) |
22 | eqidd 2827 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸)) | |
23 | eqidd 2827 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
24 | 15, 22, 23 | fmptcos 6649 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶)) |
25 | nfv 2015 | . . . . . 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 475 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) |
30 | 25, 27, 14, 29 | csbiedf 3779 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ⦋𝐸 / 𝑥⦌𝐶 = 𝐻) |
31 | 30 | mpteq2dva 4968 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
32 | 24, 31 | eqtrd 2862 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
33 | 32 | oveq2d 6922 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸))) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
34 | 21, 33 | eqtrd 2862 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Ⅎwnfc 2957 ∀wral 3118 ∃!wreu 3120 Vcvv 3415 ⦋csb 3758 ⊆ wss 3799 ↦ cmpt 4953 ∘ ccom 5347 –1-1-onto→wf1o 6123 ‘cfv 6124 (class class class)co 6906 Fincfn 8223 Basecbs 16223 0gc0g 16454 Σg cgsu 16455 CMndccmn 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-fzo 12762 df-seq 13097 df-hash 13412 df-0g 16456 df-gsum 16457 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-cntz 18101 df-cmn 18549 |
This theorem is referenced by: gsummpt2co 30326 mdetpmtr1 30435 |
Copyright terms: Public domain | W3C validator |