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Mirrors > Home > MPE Home > Th. List > dpjeq | Structured version Visualization version GIF version |
Description: Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjfval.p | ⊢ 𝑃 = (𝐺dProj𝑆) |
dpjidcl.3 | ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
dpjidcl.0 | ⊢ 0 = (0g‘𝐺) |
dpjidcl.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
dpjeq.c | ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊) |
Ref | Expression |
---|---|
dpjeq | ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpjfval.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dpjfval.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
4 | dpjidcl.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) | |
5 | dpjidcl.0 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
6 | dpjidcl.w | . . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
7 | 1, 2, 3, 4, 5, 6 | dpjidcl 19706 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |
8 | 7 | simprd 497 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
9 | 8 | eqeq1d 2738 | . 2 ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)))) |
10 | 7 | simpld 496 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊) |
11 | dpjeq.c | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊) | |
12 | 5, 6, 1, 2, 10, 11 | dprdf11 19671 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶))) |
13 | fvex 6817 | . . . 4 ⊢ ((𝑃‘𝑥)‘𝐴) ∈ V | |
14 | 13 | rgenw 3066 | . . 3 ⊢ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) ∈ V |
15 | mpteqb 6926 | . . 3 ⊢ (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) ∈ V → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶)) | |
16 | 14, 15 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶)) |
17 | 9, 12, 16 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ∀wral 3062 {crab 3284 Vcvv 3437 class class class wbr 5081 ↦ cmpt 5164 dom cdm 5600 ‘cfv 6458 (class class class)co 7307 Xcixp 8716 finSupp cfsupp 9172 0gc0g 17195 Σg cgsu 17196 DProd cdprd 19641 dProjcdpj 19642 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-tpos 8073 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-seq 13768 df-hash 14091 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-0g 17197 df-gsum 17198 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-mhm 18475 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-mulg 18746 df-subg 18797 df-ghm 18877 df-gim 18920 df-cntz 18968 df-oppg 18995 df-lsm 19286 df-pj1 19287 df-cmn 19433 df-dprd 19643 df-dpj 19644 |
This theorem is referenced by: dpjrid 19710 dchrptlem3 26459 |
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