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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 6717 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
2 | 1 | fveq1d 6719 | . . . 4 ⊢ (𝑥 = 𝑌 → ((𝑃‘𝑥)‘𝐴) = ((𝑃‘𝑌)‘𝐴)) |
3 | eqeq1 2741 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑋 ↔ 𝑌 = 𝑋)) | |
4 | 3 | ifbid 4462 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑥 = 𝑋, 𝐴, 0 ) = if(𝑌 = 𝑋, 𝐴, 0 )) |
5 | 2, 4 | eqeq12d 2753 | . . 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 2737 | . . . . . . 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 2737 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dprdfid 19404 | . . . . . 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 499 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴) |
15 | 14 | eqcomd 2743 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )))) |
16 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
17 | 8, 9, 10 | dprdub 19412 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) |
18 | 17, 11 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
19 | 13 | simpld 498 | . . . . 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 19446 | . . . 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 235 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 )) |
22 | dpjrid.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
23 | 5, 21, 22 | rspcdva 3539 | . 2 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 )) |
24 | dpjrid.6 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋) | |
25 | ifnefalse 4451 | . . 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 2777 | 1 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 {crab 3065 ifcif 4439 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 Xcixp 8578 finSupp cfsupp 8985 0gc0g 16944 Σg cgsu 16945 DProd cdprd 19380 dProjcdpj 19381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-gim 18663 df-cntz 18711 df-oppg 18738 df-lsm 19025 df-pj1 19026 df-cmn 19172 df-dprd 19382 df-dpj 19383 |
This theorem is referenced by: (None) |
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