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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 18946 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋))) |
12 | 1, 11 | mpdan 687 | . . 3 ⊢ (𝜑 → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋))) |
13 | 12 | eqeq1d 2741 | . 2 ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (𝐵 + 𝐶))) |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj1f 18944 | . . . 4 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
15 | 14, 1 | ffvelrnd 6865 | . . 3 ⊢ (𝜑 → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇) |
16 | pj1eq.6 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑇) | |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj2f 18945 | . . . 4 ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈) |
18 | 17, 1 | ffvelrnd 6865 | . . 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 18940 | . 2 ⊢ (𝜑 → ((((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) |
21 | 13, 20 | bitrd 282 | 1 ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3843 ⊆ wss 3844 {csn 4517 ‘cfv 6340 (class class class)co 7173 +gcplusg 16671 0gc0g 16819 SubGrpcsubg 18394 Cntzccntz 18566 LSSumclsm 18880 proj1cpj1 18881 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-0g 16821 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-cntz 18568 df-lsm 18882 df-pj1 18883 |
This theorem is referenced by: pj1lid 18948 pj1rid 18949 pj1ghm 18950 lsmhash 18952 dpjidcl 19302 pj1lmhm 19994 |
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