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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 6893 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
2 | 1 | fveq1d 6895 | . . . 4 ⊢ (𝑥 = 𝑌 → ((𝑃‘𝑥)‘𝐴) = ((𝑃‘𝑌)‘𝐴)) |
3 | eqeq1 2730 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑋 ↔ 𝑌 = 𝑋)) | |
4 | 3 | ifbid 4546 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑥 = 𝑋, 𝐴, 0 ) = if(𝑌 = 𝑋, 𝐴, 0 )) |
5 | 2, 4 | eqeq12d 2742 | . . 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 2726 | . . . . . . 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 2726 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dprdfid 20013 | . . . . . 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 494 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴) |
15 | 14 | eqcomd 2732 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )))) |
16 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
17 | 8, 9, 10 | dprdub 20021 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) |
18 | 17, 11 | sseldd 3979 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
19 | 13 | simpld 493 | . . . . 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 20055 | . . . 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 231 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 )) |
22 | dpjrid.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
23 | 5, 21, 22 | rspcdva 3608 | . 2 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 )) |
24 | dpjrid.6 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋) | |
25 | ifnefalse 4535 | . . 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 2766 | 1 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 {crab 3419 ifcif 4523 class class class wbr 5145 ↦ cmpt 5228 dom cdm 5674 ‘cfv 6546 (class class class)co 7416 Xcixp 8918 finSupp cfsupp 9398 0gc0g 17449 Σg cgsu 17450 DProd cdprd 19989 dProjcdpj 19990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-seq 14016 df-hash 14343 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-0g 17451 df-gsum 17452 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-gim 19249 df-cntz 19307 df-oppg 19336 df-lsm 19630 df-pj1 19631 df-cmn 19776 df-dprd 19991 df-dpj 19992 |
This theorem is referenced by: (None) |
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