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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 6672 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
2 | 1 | fveq1d 6674 | . . . 4 ⊢ (𝑥 = 𝑌 → ((𝑃‘𝑥)‘𝐴) = ((𝑃‘𝑌)‘𝐴)) |
3 | eqeq1 2827 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑋 ↔ 𝑌 = 𝑋)) | |
4 | 3 | ifbid 4491 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑥 = 𝑋, 𝐴, 0 ) = if(𝑌 = 𝑋, 𝐴, 0 )) |
5 | 2, 4 | eqeq12d 2839 | . . 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 2823 | . . . . . . 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 2823 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dprdfid 19141 | . . . . . 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 498 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴) |
15 | 14 | eqcomd 2829 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )))) |
16 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
17 | 8, 9, 10 | dprdub 19149 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) |
18 | 17, 11 | sseldd 3970 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
19 | 13 | simpld 497 | . . . . 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 19183 | . . . 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 234 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 )) |
22 | dpjrid.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
23 | 5, 21, 22 | rspcdva 3627 | . 2 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 )) |
24 | dpjrid.6 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋) | |
25 | ifnefalse 4481 | . . 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 2858 | 1 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 finSupp cfsupp 8835 0gc0g 16715 Σg cgsu 16716 DProd cdprd 19117 dProjcdpj 19118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-pj1 18764 df-cmn 18910 df-dprd 19119 df-dpj 19120 |
This theorem is referenced by: (None) |
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