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Mirrors > Home > HSE Home > Th. List > pjclem4a | Structured version Visualization version GIF version |
Description: Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjclem1.1 | ⊢ 𝐺 ∈ Cℋ |
pjclem1.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjclem4a | ⊢ (𝐴 ∈ (𝐺 ∩ 𝐻) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3965 | . 2 ⊢ (𝐴 ∈ (𝐺 ∩ 𝐻) ↔ (𝐴 ∈ 𝐺 ∧ 𝐴 ∈ 𝐻)) | |
2 | pjclem1.2 | . . . . . 6 ⊢ 𝐻 ∈ Cℋ | |
3 | 2 | cheli 31070 | . . . . 5 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) |
4 | 3 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
5 | pjclem1.1 | . . . . 5 ⊢ 𝐺 ∈ Cℋ | |
6 | 5, 2 | pjcoi 31996 | . . . 4 ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴))) |
7 | 4, 6 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐴 ∈ 𝐻) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴))) |
8 | pjid 31533 | . . . . . 6 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → ((projℎ‘𝐻)‘𝐴) = 𝐴) | |
9 | 2, 8 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ 𝐻 → ((projℎ‘𝐻)‘𝐴) = 𝐴) |
10 | eleq1 2817 | . . . . . . 7 ⊢ (((projℎ‘𝐻)‘𝐴) = 𝐴 → (((projℎ‘𝐻)‘𝐴) ∈ 𝐺 ↔ 𝐴 ∈ 𝐺)) | |
11 | pjid 31533 | . . . . . . . 8 ⊢ ((𝐺 ∈ Cℋ ∧ ((projℎ‘𝐻)‘𝐴) ∈ 𝐺) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴)) | |
12 | 5, 11 | mpan 688 | . . . . . . 7 ⊢ (((projℎ‘𝐻)‘𝐴) ∈ 𝐺 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴)) |
13 | 10, 12 | syl6bir 253 | . . . . . 6 ⊢ (((projℎ‘𝐻)‘𝐴) = 𝐴 → (𝐴 ∈ 𝐺 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴))) |
14 | eqeq2 2740 | . . . . . 6 ⊢ (((projℎ‘𝐻)‘𝐴) = 𝐴 → (((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = 𝐴)) | |
15 | 13, 14 | sylibd 238 | . . . . 5 ⊢ (((projℎ‘𝐻)‘𝐴) = 𝐴 → (𝐴 ∈ 𝐺 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = 𝐴)) |
16 | 9, 15 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝐻 → (𝐴 ∈ 𝐺 → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = 𝐴)) |
17 | 16 | impcom 406 | . . 3 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐴 ∈ 𝐻) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)) = 𝐴) |
18 | 7, 17 | eqtrd 2768 | . 2 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐴 ∈ 𝐻) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = 𝐴) |
19 | 1, 18 | sylbi 216 | 1 ⊢ (𝐴 ∈ (𝐺 ∩ 𝐻) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3948 ∘ ccom 5686 ‘cfv 6553 ℋchba 30757 Cℋ cch 30767 projℎcpjh 30775 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cc 10468 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 ax-hilex 30837 ax-hfvadd 30838 ax-hvcom 30839 ax-hvass 30840 ax-hv0cl 30841 ax-hvaddid 30842 ax-hfvmul 30843 ax-hvmulid 30844 ax-hvmulass 30845 ax-hvdistr1 30846 ax-hvdistr2 30847 ax-hvmul0 30848 ax-hfi 30917 ax-his1 30920 ax-his2 30921 ax-his3 30922 ax-his4 30923 ax-hcompl 31040 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-acn 9975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-rlim 15475 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-mulg 19038 df-cntz 19282 df-cmn 19751 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-cn 23159 df-cnp 23160 df-lm 23161 df-haus 23247 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-xms 24254 df-ms 24255 df-tms 24256 df-cfil 25211 df-cau 25212 df-cmet 25213 df-grpo 30331 df-gid 30332 df-ginv 30333 df-gdiv 30334 df-ablo 30383 df-vc 30397 df-nv 30430 df-va 30433 df-ba 30434 df-sm 30435 df-0v 30436 df-vs 30437 df-nmcv 30438 df-ims 30439 df-dip 30539 df-ssp 30560 df-ph 30651 df-cbn 30701 df-hnorm 30806 df-hba 30807 df-hvsub 30809 df-hlim 30810 df-hcau 30811 df-sh 31045 df-ch 31059 df-oc 31090 df-ch0 31091 df-shs 31146 df-pjh 31233 |
This theorem is referenced by: pjclem4 32037 pj3lem1 32044 |
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