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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 6907 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
2 | 1 | fveq1d 6909 | . . . 4 ⊢ (𝑥 = 𝑌 → ((𝑃‘𝑥)‘𝐴) = ((𝑃‘𝑌)‘𝐴)) |
3 | eqeq1 2739 | . . . . 5 ⊢ (𝑥 = 𝑌 → (𝑥 = 𝑋 ↔ 𝑌 = 𝑋)) | |
4 | 3 | ifbid 4554 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑥 = 𝑋, 𝐴, 0 ) = if(𝑌 = 𝑋, 𝐴, 0 )) |
5 | 2, 4 | eqeq12d 2751 | . . 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 2735 | . . . . . . 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 2735 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dprdfid 20052 | . . . . . 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 495 | . . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))) = 𝐴) |
15 | 14 | eqcomd 2741 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 )))) |
16 | dpjfval.p | . . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆) | |
17 | 8, 9, 10 | dprdub 20060 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) |
18 | 17, 11 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
19 | 13 | simpld 494 | . . . . 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 20094 | . . . 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 232 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = if(𝑥 = 𝑋, 𝐴, 0 )) |
22 | dpjrid.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
23 | 5, 21, 22 | rspcdva 3623 | . 2 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = if(𝑌 = 𝑋, 𝐴, 0 )) |
24 | dpjrid.6 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑋) | |
25 | ifnefalse 4543 | . . 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 2775 | 1 ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Xcixp 8936 finSupp cfsupp 9399 0gc0g 17486 Σg cgsu 17487 DProd cdprd 20028 dProjcdpj 20029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-pj1 19670 df-cmn 19815 df-dprd 20030 df-dpj 20031 |
This theorem is referenced by: (None) |
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