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Mirrors > Home > MPE Home > Th. List > dprdf11 | Structured version Visualization version GIF version |
Description: Two group sums over a direct product that give the same value are equal as functions. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
eldprdi.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
dprdf11.4 | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
Ref | Expression |
---|---|
dprdf11 | ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldprdi.w | . . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
2 | eldprdi.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
3 | eldprdi.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
4 | eldprdi.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
5 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 1, 2, 3, 4, 5 | dprdff 19353 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
7 | 6 | ffnd 6524 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | dprdf11.4 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
9 | 1, 2, 3, 8, 5 | dprdff 19353 | . . . 4 ⊢ (𝜑 → 𝐻:𝐼⟶(Base‘𝐺)) |
10 | 9 | ffnd 6524 | . . 3 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
11 | eqfnfv 6830 | . . 3 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐻 Fn 𝐼) → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) | |
12 | 7, 10, 11 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
13 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
14 | eqid 2736 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 13, 1, 2, 3, 4, 8, 14 | dprdfsub 19362 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)))) |
16 | 15 | simpld 498 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) ∈ 𝑊) |
17 | 13, 1, 2, 3, 16 | dprdfeq0 19363 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
18 | 15 | simprd 499 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻))) |
19 | 18 | eqeq1d 2738 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 )) |
20 | 2, 3 | dprddomcld 19342 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
21 | fvexd 6710 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
22 | fvexd 6710 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ V) | |
23 | 6 | feqmptd 6758 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
24 | 9 | feqmptd 6758 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐼 ↦ (𝐻‘𝑥))) |
25 | 20, 21, 22, 23, 24 | offval2 7466 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)))) |
26 | 25 | eqeq1d 2738 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
27 | ovex 7224 | . . . . . . 7 ⊢ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V | |
28 | 27 | rgenw 3063 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V |
29 | mpteqb 6815 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 )) | |
30 | 28, 29 | ax-mp 5 | . . . . 5 ⊢ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ) |
31 | dprdgrp 19346 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
32 | 2, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp) |
33 | 32 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) |
34 | 6 | ffvelrnda 6882 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺)) |
35 | 9 | ffvelrnda 6882 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ (Base‘𝐺)) |
36 | 5, 13, 14 | grpsubeq0 18403 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺) ∧ (𝐻‘𝑥) ∈ (Base‘𝐺)) → (((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐻‘𝑥))) |
37 | 33, 34, 35, 36 | syl3anc 1373 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐻‘𝑥))) |
38 | 37 | ralbidva 3107 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
39 | 30, 38 | syl5bb 286 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
40 | 26, 39 | bitrd 282 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
41 | 17, 19, 40 | 3bitr3d 312 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
42 | 5 | dprdssv 19357 | . . . 4 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
43 | 13, 1, 2, 3, 4 | eldprdi 19359 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
44 | 42, 43 | sseldi 3885 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (Base‘𝐺)) |
45 | 13, 1, 2, 3, 8 | eldprdi 19359 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (𝐺 DProd 𝑆)) |
46 | 42, 45 | sseldi 3885 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (Base‘𝐺)) |
47 | 5, 13, 14 | grpsubeq0 18403 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝐺 Σg 𝐹) ∈ (Base‘𝐺) ∧ (𝐺 Σg 𝐻) ∈ (Base‘𝐺)) → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
48 | 32, 44, 46, 47 | syl3anc 1373 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
49 | 12, 41, 48 | 3bitr2rd 311 | 1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 {crab 3055 Vcvv 3398 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 Xcixp 8556 finSupp cfsupp 8963 Basecbs 16666 0gc0g 16898 Σg cgsu 16899 Grpcgrp 18319 -gcsg 18321 DProd cdprd 19334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-gim 18617 df-cntz 18665 df-oppg 18692 df-cmn 19126 df-dprd 19336 |
This theorem is referenced by: dmdprdsplitlem 19378 dpjeq 19400 |
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