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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 2823 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 1, 2, 3, 4, 5 | dprdff 19136 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
7 | 6 | ffnd 6517 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | dprdf11.4 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
9 | 1, 2, 3, 8, 5 | dprdff 19136 | . . . 4 ⊢ (𝜑 → 𝐻:𝐼⟶(Base‘𝐺)) |
10 | 9 | ffnd 6517 | . . 3 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
11 | eqfnfv 6804 | . . 3 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐻 Fn 𝐼) → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) | |
12 | 7, 10, 11 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
13 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
14 | eqid 2823 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 13, 1, 2, 3, 4, 8, 14 | dprdfsub 19145 | . . . . 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 497 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) ∈ 𝑊) |
17 | 13, 1, 2, 3, 16 | dprdfeq0 19146 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
18 | 15 | simprd 498 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻))) |
19 | 18 | eqeq1d 2825 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 )) |
20 | 2, 3 | dprddomcld 19125 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
21 | fvexd 6687 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
22 | fvexd 6687 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ V) | |
23 | 6 | feqmptd 6735 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
24 | 9 | feqmptd 6735 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐼 ↦ (𝐻‘𝑥))) |
25 | 20, 21, 22, 23, 24 | offval2 7428 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)))) |
26 | 25 | eqeq1d 2825 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
27 | ovex 7191 | . . . . . . 7 ⊢ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V | |
28 | 27 | rgenw 3152 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V |
29 | mpteqb 6789 | . . . . . 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 19129 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
32 | 2, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp) |
33 | 32 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) |
34 | 6 | ffvelrnda 6853 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺)) |
35 | 9 | ffvelrnda 6853 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ (Base‘𝐺)) |
36 | 5, 13, 14 | grpsubeq0 18187 | . . . . . . 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 1367 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐻‘𝑥))) |
38 | 37 | ralbidva 3198 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
39 | 30, 38 | syl5bb 285 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
40 | 26, 39 | bitrd 281 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
41 | 17, 19, 40 | 3bitr3d 311 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
42 | 5 | dprdssv 19140 | . . . 4 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
43 | 13, 1, 2, 3, 4 | eldprdi 19142 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
44 | 42, 43 | sseldi 3967 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (Base‘𝐺)) |
45 | 13, 1, 2, 3, 8 | eldprdi 19142 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (𝐺 DProd 𝑆)) |
46 | 42, 45 | sseldi 3967 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (Base‘𝐺)) |
47 | 5, 13, 14 | grpsubeq0 18187 | . . 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 1367 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
49 | 12, 41, 48 | 3bitr2rd 310 | 1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 Xcixp 8463 finSupp cfsupp 8835 Basecbs 16485 0gc0g 16715 Σg cgsu 16716 Grpcgrp 18105 -gcsg 18107 DProd cdprd 19117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-cmn 18910 df-dprd 19119 |
This theorem is referenced by: dmdprdsplitlem 19161 dpjeq 19183 |
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