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Mirrors > Home > MPE Home > Th. List > pj1rid | Structured version Visualization version GIF version |
Description: The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-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) |
Ref | Expression |
---|---|
pj1rid | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1eu.2 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
2 | 1 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ (SubGrp‘𝐺)) |
3 | subgrcl 18222 | . . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐺 ∈ Grp) |
5 | pj1eu.3 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
6 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | 6 | subgss 18218 | . . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
8 | 5, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺)) |
9 | 8 | sselda 3964 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝐺)) |
10 | pj1eu.a | . . . . . 6 ⊢ + = (+g‘𝐺) | |
11 | pj1eu.o | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
12 | 6, 10, 11 | grplid 18071 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺)) → ( 0 + 𝑋) = 𝑋) |
13 | 4, 9, 12 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ( 0 + 𝑋) = 𝑋) |
14 | 13 | eqcomd 2824 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ( 0 + 𝑋)) |
15 | pj1eu.s | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
16 | pj1eu.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
17 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ (SubGrp‘𝐺)) |
18 | pj1eu.4 | . . . . 5 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑇 ∩ 𝑈) = { 0 }) |
20 | pj1eu.5 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
21 | 20 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑇 ⊆ (𝑍‘𝑈)) |
22 | pj1f.p | . . . 4 ⊢ 𝑃 = (proj1‘𝐺) | |
23 | 15 | lsmub2 18712 | . . . . . 6 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈)) |
24 | 1, 5, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊆ (𝑇 ⊕ 𝑈)) |
25 | 24 | sselda 3964 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (𝑇 ⊕ 𝑈)) |
26 | 11 | subg0cl 18225 | . . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇) |
27 | 2, 26 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 0 ∈ 𝑇) |
28 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
29 | 10, 15, 11, 16, 2, 17, 19, 21, 22, 25, 27, 28 | pj1eq 18755 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑋 = ( 0 + 𝑋) ↔ (((𝑇𝑃𝑈)‘𝑋) = 0 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝑋))) |
30 | 14, 29 | mpbid 233 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) = 0 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝑋)) |
31 | 30 | simpld 495 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 {csn 4557 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 SubGrpcsubg 18211 Cntzccntz 18383 LSSumclsm 18688 proj1cpj1 18689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-pj1 18691 |
This theorem is referenced by: dpjidcl 19109 |
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