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Mirrors > Home > HSE Home > Th. List > pjss2coi | Structured version Visualization version GIF version |
Description: Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjco.1 | ⊢ 𝐺 ∈ Cℋ |
pjco.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjss2coi | ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjco.1 | . . . . . . 7 ⊢ 𝐺 ∈ Cℋ | |
2 | pjco.2 | . . . . . . 7 ⊢ 𝐻 ∈ Cℋ | |
3 | 1, 2 | pjcoi 29017 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
4 | 3 | adantl 482 | . . . . 5 ⊢ ((𝐺 ⊆ 𝐻 ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
5 | fveq2 6191 | . . . . . . . . . 10 ⊢ (𝑥 = if(𝑥 ∈ ℋ, 𝑥, 0ℎ) → ((projℎ‘𝐻)‘𝑥) = ((projℎ‘𝐻)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) | |
6 | 5 | fveq2d 6195 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ ℋ, 𝑥, 0ℎ) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ)))) |
7 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ ℋ, 𝑥, 0ℎ) → ((projℎ‘𝐺)‘𝑥) = ((projℎ‘𝐺)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) | |
8 | 6, 7 | eqeq12d 2637 | . . . . . . . 8 ⊢ (𝑥 = if(𝑥 ∈ ℋ, 𝑥, 0ℎ) → (((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = ((projℎ‘𝐺)‘𝑥) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) = ((projℎ‘𝐺)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ)))) |
9 | 8 | imbi2d 330 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ ℋ, 𝑥, 0ℎ) → ((𝐺 ⊆ 𝐻 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = ((projℎ‘𝐺)‘𝑥)) ↔ (𝐺 ⊆ 𝐻 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) = ((projℎ‘𝐺)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))))) |
10 | ifhvhv0 27879 | . . . . . . . 8 ⊢ if(𝑥 ∈ ℋ, 𝑥, 0ℎ) ∈ ℋ | |
11 | 1, 10, 2 | pjss2i 28539 | . . . . . . 7 ⊢ (𝐺 ⊆ 𝐻 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) = ((projℎ‘𝐺)‘if(𝑥 ∈ ℋ, 𝑥, 0ℎ))) |
12 | 9, 11 | dedth 4139 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = ((projℎ‘𝐺)‘𝑥))) |
13 | 12 | impcom 446 | . . . . 5 ⊢ ((𝐺 ⊆ 𝐻 ∧ 𝑥 ∈ ℋ) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = ((projℎ‘𝐺)‘𝑥)) |
14 | 4, 13 | eqtrd 2656 | . . . 4 ⊢ ((𝐺 ⊆ 𝐻 ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘𝑥)) |
15 | 14 | ralrimiva 2966 | . . 3 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘𝑥)) |
16 | 1 | pjfi 28563 | . . . . 5 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
17 | 2 | pjfi 28563 | . . . . 5 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
18 | 16, 17 | hocofi 28625 | . . . 4 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)): ℋ⟶ ℋ |
19 | 18, 16 | hoeqi 28620 | . . 3 ⊢ (∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘𝑥) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) |
20 | 15, 19 | sylib 208 | . 2 ⊢ (𝐺 ⊆ 𝐻 → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) |
21 | fveq1 6190 | . . . . . . . . . . . 12 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑦) = ((projℎ‘𝐺)‘𝑦)) | |
22 | 21 | oveq2d 6666 | . . . . . . . . . . 11 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → (𝑥 ·ih (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑦)) = (𝑥 ·ih ((projℎ‘𝐺)‘𝑦))) |
23 | 22 | ad2antlr 763 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ ℋ ∧ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑦)) = (𝑥 ·ih ((projℎ‘𝐺)‘𝑦))) |
24 | 2, 1 | pjadjcoi 29020 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑦))) |
25 | 24 | adantlr 751 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ ℋ ∧ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) ∧ 𝑦 ∈ ℋ) → ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑦))) |
26 | 1 | pjadji 28544 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((projℎ‘𝐺)‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((projℎ‘𝐺)‘𝑦))) |
27 | 26 | adantlr 751 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ ℋ ∧ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) ∧ 𝑦 ∈ ℋ) → (((projℎ‘𝐺)‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((projℎ‘𝐺)‘𝑦))) |
28 | 23, 25, 27 | 3eqtr4d 2666 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℋ ∧ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) ∧ 𝑦 ∈ ℋ) → ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (((projℎ‘𝐺)‘𝑥) ·ih 𝑦)) |
29 | 28 | exp31 630 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → (𝑦 ∈ ℋ → ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (((projℎ‘𝐺)‘𝑥) ·ih 𝑦)))) |
30 | 29 | ralrimdv 2968 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → ∀𝑦 ∈ ℋ ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (((projℎ‘𝐺)‘𝑥) ·ih 𝑦))) |
31 | 2, 1 | pjcohcli 29019 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ∈ ℋ) |
32 | 1 | pjhcli 28277 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐺)‘𝑥) ∈ ℋ) |
33 | hial2eq 27963 | . . . . . . . 8 ⊢ (((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ∈ ℋ ∧ ((projℎ‘𝐺)‘𝑥) ∈ ℋ) → (∀𝑦 ∈ ℋ ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (((projℎ‘𝐺)‘𝑥) ·ih 𝑦) ↔ (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥))) | |
34 | 31, 32, 33 | syl2anc 693 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (∀𝑦 ∈ ℋ ((((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) ·ih 𝑦) = (((projℎ‘𝐺)‘𝑥) ·ih 𝑦) ↔ (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥))) |
35 | 30, 34 | sylibd 229 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥))) |
36 | 35 | com12 32 | . . . . 5 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → (𝑥 ∈ ℋ → (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥))) |
37 | 36 | ralrimiv 2965 | . . . 4 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → ∀𝑥 ∈ ℋ (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥)) |
38 | 17, 16 | hocofi 28625 | . . . . 5 ⊢ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)): ℋ⟶ ℋ |
39 | 38, 16 | hoeqi 28620 | . . . 4 ⊢ (∀𝑥 ∈ ℋ (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝑥) = ((projℎ‘𝐺)‘𝑥) ↔ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺)) |
40 | 37, 39 | sylib 208 | . . 3 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺)) |
41 | 1, 2 | pjss1coi 29022 | . . 3 ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺)) |
42 | 40, 41 | sylibr 224 | . 2 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺) → 𝐺 ⊆ 𝐻) |
43 | 20, 42 | impbii 199 | 1 ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ifcif 4086 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ·ih csp 27779 0ℎc0v 27781 Cℋ cch 27786 projℎcpjh 27794 |
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-inf2 8538 ax-cc 9257 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 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-lm 21033 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cfil 23053 df-cau 23054 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-ssp 27577 df-ph 27668 df-cbn 27719 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 df-hcau 27830 df-sh 28064 df-ch 28078 df-oc 28109 df-ch0 28110 df-shs 28167 df-pjh 28254 |
This theorem is referenced by: pjidmcoi 29036 pjin2i 29052 pjin3i 29053 |
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