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Mirrors > Home > MPE Home > Th. List > dpjlid | Structured version Visualization version GIF version |
Description: The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (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)) |
Ref | Expression |
---|---|
dpjlid | ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpjfval.1 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dpjfval.2 | . . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | dpjfval.p | . . . 4 ⊢ 𝑃 = (𝐺dProj𝑆) | |
4 | eqid 2740 | . . . 4 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
5 | dpjlid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
6 | 1, 2, 3, 4, 5 | dpjval 19657 | . . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
7 | 6 | fveq1d 6773 | . 2 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴)) |
8 | dpjlid.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) | |
9 | eqid 2740 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2740 | . . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
11 | eqid 2740 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | eqid 2740 | . . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
13 | 1, 2 | dprdf2 19608 | . . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
14 | 13, 5 | ffvelrnd 6959 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
15 | difssd 4072 | . . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
16 | 1, 2, 15 | dprdres 19629 | . . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
17 | 16 | simpld 495 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
18 | dprdsubg 19625 | . . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
20 | 1, 2, 5, 11 | dpjdisj 19654 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
21 | 1, 2, 5, 12 | dpjcntz 19653 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 9, 10, 11, 12, 14, 19, 20, 21, 4 | pj1lid 19305 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴) |
23 | 8, 22 | mpdan 684 | . 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴) |
24 | 7, 23 | eqtrd 2780 | 1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ⊆ wss 3892 {csn 4567 class class class wbr 5079 dom cdm 5590 ↾ cres 5592 ‘cfv 6432 (class class class)co 7271 +gcplusg 16960 0gc0g 17148 SubGrpcsubg 18747 Cntzccntz 18919 LSSumclsm 19237 proj1cpj1 19238 DProd cdprd 19594 dProjcdpj 19595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-gim 18873 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-pj1 19240 df-cmn 19386 df-dprd 19596 df-dpj 19597 |
This theorem is referenced by: (None) |
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