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| Mirrors > Home > MPE Home > Th. List > Mathboxes > gsummptfsf1o | Structured version Visualization version GIF version | ||
| Description: Re-index a finite group sum using a bijection. A version of gsummptf1o 19981 expressed using finite support. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| Ref | Expression |
|---|---|
| gsummptfsf1o.x | ⊢ Ⅎ𝑥𝐻 |
| gsummptfsf1o.b | ⊢ 𝐵 = (Base‘𝐺) |
| gsummptfsf1o.z | ⊢ 0 = (0g‘𝐺) |
| gsummptfsf1o.i | ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) |
| gsummptfsf1o.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| gsummptfsf1o.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| gsummptfsf1o.a | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
| gsummptfsf1o.d | ⊢ (𝜑 → 𝐹 ⊆ 𝐵) |
| gsummptfsf1o.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) |
| gsummptfsf1o.e | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) |
| gsummptfsf1o.h | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
| Ref | Expression |
|---|---|
| gsummptfsf1o | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gsummptfsf1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | gsummptfsf1o.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 3 | gsummptfsf1o.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 4 | gsummptfsf1o.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | gsummptfsf1o.d | . . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝐵) | |
| 6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ⊆ 𝐵) |
| 7 | gsummptfsf1o.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) | |
| 8 | 6, 7 | sseldd 3984 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
| 9 | 8 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
| 10 | gsummptfsf1o.a | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) | |
| 11 | gsummptfsf1o.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) | |
| 12 | 11 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴) |
| 13 | gsummptfsf1o.h | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) | |
| 14 | 13 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
| 15 | eqid 2737 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸) | |
| 16 | 15 | f1ompt 7131 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴 ↔ (∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)) |
| 17 | 12, 14, 16 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴) |
| 18 | 1, 2, 3, 4, 9, 10, 17 | gsumf1o 19934 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)))) |
| 19 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸)) | |
| 20 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
| 21 | 12, 19, 20 | fmptcos 7151 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶)) |
| 22 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐷) | |
| 23 | gsummptfsf1o.x | . . . . . . 7 ⊢ Ⅎ𝑥𝐻 | |
| 24 | 23 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → Ⅎ𝑥𝐻) |
| 25 | gsummptfsf1o.i | . . . . . . 7 ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) | |
| 26 | 25 | adantl 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) |
| 27 | 22, 24, 11, 26 | csbiedf 3929 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ⦋𝐸 / 𝑥⦌𝐶 = 𝐻) |
| 28 | 27 | mpteq2dva 5242 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
| 29 | 21, 28 | eqtrd 2777 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
| 30 | 29 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸))) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
| 31 | 18, 30 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃!wreu 3378 ⦋csb 3899 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ∘ ccom 5689 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: (None) |
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