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Mirrors > Home > MPE Home > Th. List > pj1eq | Structured version Visualization version GIF version |
Description: Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pj1eu.a | ⊢ + = (+g‘𝐺) |
pj1eu.s | ⊢ ⊕ = (LSSum‘𝐺) |
pj1eu.o | ⊢ 0 = (0g‘𝐺) |
pj1eu.z | ⊢ 𝑍 = (Cntz‘𝐺) |
pj1eu.2 | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
pj1eu.3 | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
pj1eu.4 | ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
pj1eu.5 | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
pj1f.p | ⊢ 𝑃 = (proj1‘𝐺) |
pj1eq.5 | ⊢ (𝜑 → 𝑋 ∈ (𝑇 ⊕ 𝑈)) |
pj1eq.6 | ⊢ (𝜑 → 𝐵 ∈ 𝑇) |
pj1eq.7 | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
pj1eq | ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1eq.5 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑇 ⊕ 𝑈)) | |
2 | pj1eu.a | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | pj1eu.s | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
4 | pj1eu.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
5 | pj1eu.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝐺) | |
6 | pj1eu.2 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
7 | pj1eu.3 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
8 | pj1eu.4 | . . . . 5 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
9 | pj1eu.5 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
10 | pj1f.p | . . . . 5 ⊢ 𝑃 = (proj1‘𝐺) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj1id 18112 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋))) |
12 | 1, 11 | mpdan 702 | . . 3 ⊢ (𝜑 → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋))) |
13 | 12 | eqeq1d 2624 | . 2 ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (𝐵 + 𝐶))) |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj1f 18110 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
15 | 14, 1 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇) |
16 | pj1eq.6 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑇) | |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj2f 18111 | . . . 4 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈) |
18 | 17, 1 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → ((𝑈𝑃𝑇)‘𝑋) ∈ 𝑈) |
19 | pj1eq.7 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
20 | 2, 4, 5, 6, 7, 8, 9, 15, 16, 18, 19 | subgdisjb 18106 | . 2 ⊢ (𝜑 → ((((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) |
21 | 13, 20 | bitrd 268 | 1 ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ‘cfv 5888 (class class class)co 6650 +gcplusg 15941 0gc0g 16100 SubGrpcsubg 17588 Cntzccntz 17748 LSSumclsm 18049 proj1cpj1 18050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-pj1 18052 |
This theorem is referenced by: pj1lid 18114 pj1rid 18115 pj1ghm 18116 lsmhash 18118 dpjidcl 18457 pj1lmhm 19100 |
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