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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 2731 | . . . 4 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
5 | dpjlid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
6 | 1, 2, 3, 4, 5 | dpjval 19971 | . . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
7 | 6 | fveq1d 6893 | . 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 2731 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2731 | . . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
11 | eqid 2731 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | eqid 2731 | . . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
13 | 1, 2 | dprdf2 19922 | . . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
14 | 13, 5 | ffvelcdmd 7087 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
15 | difssd 4132 | . . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
16 | 1, 2, 15 | dprdres 19943 | . . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
17 | 16 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
18 | dprdsubg 19939 | . . . . 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 19968 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
21 | 1, 2, 5, 12 | dpjcntz 19967 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 9, 10, 11, 12, 14, 19, 20, 21, 4 | pj1lid 19614 | . . 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 2771 | 1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 class class class wbr 5148 dom cdm 5676 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 +gcplusg 17204 0gc0g 17392 SubGrpcsubg 19040 Cntzccntz 19224 LSSumclsm 19547 proj1cpj1 19548 DProd cdprd 19908 dProjcdpj 19909 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-gim 19177 df-cntz 19226 df-oppg 19255 df-lsm 19549 df-pj1 19550 df-cmn 19695 df-dprd 19910 df-dpj 19911 |
This theorem is referenced by: (None) |
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