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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 19111 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | eqid 2725 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 4 | subgss 19107 | . . . 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 19107 | . . . 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 19678 | . . 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 19680 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) |
20 | riotacl 7393 | . . 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 7125 | 1 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∃!wreu 3361 ∩ cin 3943 ⊆ wss 3944 {csn 4630 ↦ cmpt 5232 ⟶wf 6545 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17199 +gcplusg 17252 0gc0g 17440 Grpcgrp 18914 SubGrpcsubg 19100 Cntzccntz 19295 LSSumclsm 19618 proj1cpj1 19619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19103 df-cntz 19297 df-lsm 19620 df-pj1 19621 |
This theorem is referenced by: pj2f 19682 pj1id 19683 pj1eq 19684 pj1ghm 19687 pj1ghm2 19688 lsmhash 19689 dpjf 20043 pj1lmhm 21014 pj1lmhm2 21015 pjdm2 21679 pjf2 21682 |
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