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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 2732 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 1, 2, 3, 4, 5 | dprdff 19843 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
7 | 6 | ffnd 6706 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
8 | dprdf11.4 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
9 | 1, 2, 3, 8, 5 | dprdff 19843 | . . . 4 ⊢ (𝜑 → 𝐻:𝐼⟶(Base‘𝐺)) |
10 | 9 | ffnd 6706 | . . 3 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
11 | eqfnfv 7019 | . . 3 ⊢ ((𝐹 Fn 𝐼 ∧ 𝐻 Fn 𝐼) → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) | |
12 | 7, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 = 𝐻 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
13 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
14 | eqid 2732 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 13, 1, 2, 3, 4, 8, 14 | dprdfsub 19852 | . . . . 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 495 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) ∈ 𝑊) |
17 | 13, 1, 2, 3, 16 | dprdfeq0 19853 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
18 | 15 | simprd 496 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻))) |
19 | 18 | eqeq1d 2734 | . . 3 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ∘f (-g‘𝐺)𝐻)) = 0 ↔ ((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 )) |
20 | 2, 3 | dprddomcld 19832 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
21 | fvexd 6894 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ V) | |
22 | fvexd 6894 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ V) | |
23 | 6 | feqmptd 6947 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
24 | 9 | feqmptd 6947 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐼 ↦ (𝐻‘𝑥))) |
25 | 20, 21, 22, 23, 24 | offval2 7674 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)))) |
26 | 25 | eqeq1d 2734 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ))) |
27 | ovex 7427 | . . . . . . 7 ⊢ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V | |
28 | 27 | rgenw 3065 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) ∈ V |
29 | mpteqb 7004 | . . . . . 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 19836 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
32 | 2, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp) |
33 | 32 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) |
34 | 6 | ffvelcdmda 7072 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺)) |
35 | 9 | ffvelcdmda 7072 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ (Base‘𝐺)) |
36 | 5, 13, 14 | grpsubeq0 18885 | . . . . . . 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 1371 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐻‘𝑥))) |
38 | 37 | ralbidva 3175 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
39 | 30, 38 | bitrid 282 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(-g‘𝐺)(𝐻‘𝑥))) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
40 | 26, 39 | bitrd 278 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f (-g‘𝐺)𝐻) = (𝑥 ∈ 𝐼 ↦ 0 ) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
41 | 17, 19, 40 | 3bitr3d 308 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) = (𝐻‘𝑥))) |
42 | 5 | dprdssv 19847 | . . . 4 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
43 | 13, 1, 2, 3, 4 | eldprdi 19849 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
44 | 42, 43 | sselid 3977 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (Base‘𝐺)) |
45 | 13, 1, 2, 3, 8 | eldprdi 19849 | . . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (𝐺 DProd 𝑆)) |
46 | 42, 45 | sselid 3977 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐻) ∈ (Base‘𝐺)) |
47 | 5, 13, 14 | grpsubeq0 18885 | . . 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 1371 | . 2 ⊢ (𝜑 → (((𝐺 Σg 𝐹)(-g‘𝐺)(𝐺 Σg 𝐻)) = 0 ↔ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
49 | 12, 41, 48 | 3bitr2rd 307 | 1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5670 Fn wfn 6528 ‘cfv 6533 (class class class)co 7394 ∘f cof 7652 Xcixp 8876 finSupp cfsupp 9346 Basecbs 17128 0gc0g 17369 Σg cgsu 17370 Grpcgrp 18796 -gcsg 18798 DProd cdprd 19824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-fzo 13612 df-seq 13951 df-hash 14275 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-0g 17371 df-gsum 17372 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mhm 18649 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-mulg 18925 df-subg 18977 df-ghm 19058 df-gim 19101 df-cntz 19149 df-oppg 19176 df-cmn 19616 df-dprd 19826 |
This theorem is referenced by: dmdprdsplitlem 19868 dpjeq 19890 |
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