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Mirrors > Home > MPE Home > Th. List > dpjrid | Structured version Visualization version GIF version |
Description: The 𝑌-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjfval.p | ⊢ 𝑃 = (𝐺dProj𝑆) |
dpjlid.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
dpjlid.4 | ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) |
dpjrid.0 | ⊢ 0 = (0g‘𝐺) |
dpjrid.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐼) |
dpjrid.6 | ⊢ (𝜑 → 𝑌 ≠ 𝑋) |
Ref | Expression |
---|---|
dpjrid | ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6437 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
2 | 1 | fveq1d 6439 | . . . 4 ⊢ (𝑥 = 𝑌 → ((𝑃‘𝑥)‘𝐴) = ((𝑃‘𝑌)‘𝐴)) |
3 | eqeq1 2829 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑋 ↔ 𝑌 = 𝑋)) | |
4 | 3 | ifbid 4330 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑥 = 𝑋, 𝐴, 0 ) = if(𝑌 = 𝑋, 𝐴, 0 )) |
5 | 2, 4 | eqeq12d 2840 | . . 3 ⊢ (𝑥 = 𝑌 → (((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 ) ↔ ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 ))) |
6 | dpjrid.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
7 | eqid 2825 | . . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
8 | dpjfval.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
9 | dpjfval.2 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
10 | dpjlid.3 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
11 | dpjlid.4 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) | |
12 | eqid 2825 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dprdfid 18777 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴)) |
14 | 13 | simprd 491 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴) |
15 | 14 | eqcomd 2831 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )))) |
16 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
17 | 8, 9, 10 | dprdub 18785 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) |
18 | 17, 11 | sseldd 3828 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
19 | 13 | simpld 490 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }) |
20 | 8, 9, 16, 18, 6, 7, 19 | dpjeq 18819 | . . . 4 ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 ))) |
21 | 15, 20 | mpbid 224 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 )) |
22 | dpjrid.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
23 | 5, 21, 22 | rspcdva 3532 | . 2 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 )) |
24 | dpjrid.6 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋) | |
25 | ifnefalse 4320 | . . 3 ⊢ (𝑌 ≠ 𝑋 → if(𝑌 = 𝑋, 𝐴, 0 ) = 0 ) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝜑 → if(𝑌 = 𝑋, 𝐴, 0 ) = 0 ) |
27 | 23, 26 | eqtrd 2861 | 1 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 {crab 3121 ifcif 4308 class class class wbr 4875 ↦ cmpt 4954 dom cdm 5346 ‘cfv 6127 (class class class)co 6910 Xcixp 8181 finSupp cfsupp 8550 0gc0g 16460 Σg cgsu 16461 DProd cdprd 18753 dProjcdpj 18754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-gsum 16463 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-gim 18059 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-pj1 18410 df-cmn 18555 df-dprd 18755 df-dpj 18756 |
This theorem is referenced by: (None) |
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