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Mirrors > Home > MPE Home > Th. List > pj1f | Structured version Visualization version GIF version |
Description: The left projection function maps a direct subspace sum onto the left factor. (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 |
---|---|
pj1f | ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1eu.2 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 18276 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 4 | subgss 18272 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
7 | pj1eu.3 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
8 | 4 | subgss 18272 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺)) |
10 | pj1eu.a | . . . 4 ⊢ + = (+g‘𝐺) | |
11 | pj1eu.s | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
12 | pj1f.p | . . . 4 ⊢ 𝑃 = (proj1‘𝐺) | |
13 | 4, 10, 11, 12 | pj1fval 18812 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))) |
14 | 3, 6, 9, 13 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))) |
15 | pj1eu.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
16 | pj1eu.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
17 | pj1eu.4 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
18 | pj1eu.5 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
19 | 10, 11, 15, 16, 1, 7, 17, 18 | pj1eu 18814 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) |
20 | riotacl 7110 | . . 3 ⊢ (∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) ∈ 𝑇) | |
21 | 19, 20 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) ∈ 𝑇) |
22 | 14, 21 | fmpt3d 6857 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∃!wreu 3108 ∩ cin 3880 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 SubGrpcsubg 18265 Cntzccntz 18437 LSSumclsm 18751 proj1cpj1 18752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-pj1 18754 |
This theorem is referenced by: pj2f 18816 pj1id 18817 pj1eq 18818 pj1ghm 18821 pj1ghm2 18822 lsmhash 18823 dpjf 19172 pj1lmhm 19865 pj1lmhm2 19866 pjdm2 20400 pjf2 20403 |
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