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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 2759 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 1, 2, 3, 4, 5 | dprdff 19187 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
7 | 6 | ffnd 6492 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | dprdf11.4 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
9 | 1, 2, 3, 8, 5 | dprdff 19187 | . . . 4 ⊢ (𝜑 → 𝐻:𝐼⟶(Base‘𝐺)) |
10 | 9 | ffnd 6492 | . . 3 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
11 | eqfnfv 6786 | . . 3 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐻 Fn 𝐼) → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) | |
12 | 7, 10, 11 | syl2anc 588 | . 2 ⊢ (𝜑 → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
13 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
14 | eqid 2759 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 13, 1, 2, 3, 4, 8, 14 | dprdfsub 19196 | . . . . 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 499 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) ∈ 𝑊) |
17 | 13, 1, 2, 3, 16 | dprdfeq0 19197 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
18 | 15 | simprd 500 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻))) |
19 | 18 | eqeq1d 2761 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 )) |
20 | 2, 3 | dprddomcld 19176 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
21 | fvexd 6666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
22 | fvexd 6666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ V) | |
23 | 6 | feqmptd 6714 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
24 | 9 | feqmptd 6714 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐼 ↦ (𝐻‘𝑥))) |
25 | 20, 21, 22, 23, 24 | offval2 7417 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)))) |
26 | 25 | eqeq1d 2761 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
27 | ovex 7176 | . . . . . . 7 ⊢ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V | |
28 | 27 | rgenw 3080 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V |
29 | mpteqb 6771 | . . . . . 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 19180 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
32 | 2, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp) |
33 | 32 | adantr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) |
34 | 6 | ffvelrnda 6835 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺)) |
35 | 9 | ffvelrnda 6835 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ (Base‘𝐺)) |
36 | 5, 13, 14 | grpsubeq0 18237 | . . . . . . 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 1369 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐻‘𝑥))) |
38 | 37 | ralbidva 3123 | . . . . 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 313 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
42 | 5 | dprdssv 19191 | . . . 4 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
43 | 13, 1, 2, 3, 4 | eldprdi 19193 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
44 | 42, 43 | sseldi 3886 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (Base‘𝐺)) |
45 | 13, 1, 2, 3, 8 | eldprdi 19193 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (𝐺 DProd 𝑆)) |
46 | 42, 45 | sseldi 3886 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (Base‘𝐺)) |
47 | 5, 13, 14 | grpsubeq0 18237 | . . 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 1369 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
49 | 12, 41, 48 | 3bitr2rd 312 | 1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∀wral 3068 {crab 3072 Vcvv 3407 class class class wbr 5025 ↦ cmpt 5105 dom cdm 5517 Fn wfn 6323 ‘cfv 6328 (class class class)co 7143 ∘f cof 7396 Xcixp 8472 finSupp cfsupp 8851 Basecbs 16526 0gc0g 16756 Σg cgsu 16757 Grpcgrp 18154 -gcsg 18156 DProd cdprd 19168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-seq 13404 df-hash 13726 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-0g 16758 df-gsum 16759 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-mhm 18007 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-mulg 18277 df-subg 18328 df-ghm 18408 df-gim 18451 df-cntz 18499 df-oppg 18526 df-cmn 18960 df-dprd 19170 |
This theorem is referenced by: dmdprdsplitlem 19212 dpjeq 19234 |
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