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Mirrors > Home > HSE Home > Th. List > pjeq | Structured version Visualization version GIF version |
Description: Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjeq | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) = 𝐵 ↔ (𝐵 ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 +ℎ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhth 31326 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) = ℋ) | |
2 | 1 | eleq2d 2812 | . . 3 ⊢ (𝐻 ∈ Cℋ → (𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ 𝐴 ∈ ℋ)) |
3 | 2 | biimpar 476 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → 𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻))) |
4 | pjpreeq 31331 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻))) → (((projℎ‘𝐻)‘𝐴) = 𝐵 ↔ (𝐵 ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 +ℎ 𝑥)))) | |
5 | 3, 4 | syldan 589 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) = 𝐵 ↔ (𝐵 ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 +ℎ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 ‘cfv 6554 (class class class)co 7424 ℋchba 30852 +ℎ cva 30853 Cℋ cch 30862 ⊥cort 30863 +ℋ cph 30864 projℎcpjh 30870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cc 10478 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 ax-hilex 30932 ax-hfvadd 30933 ax-hvcom 30934 ax-hvass 30935 ax-hv0cl 30936 ax-hvaddid 30937 ax-hfvmul 30938 ax-hvmulid 30939 ax-hvmulass 30940 ax-hvdistr1 30941 ax-hvdistr2 30942 ax-hvmul0 30943 ax-hfi 31012 ax-his1 31015 ax-his2 31016 ax-his3 31017 ax-his4 31018 ax-hcompl 31135 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-acn 9985 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ico 13384 df-icc 13385 df-fz 13539 df-fl 13812 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-rest 17437 df-topgen 17458 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-top 22887 df-topon 22904 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lm 23224 df-haus 23310 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-cfil 25274 df-cau 25275 df-cmet 25276 df-grpo 30426 df-gid 30427 df-ginv 30428 df-gdiv 30429 df-ablo 30478 df-vc 30492 df-nv 30525 df-va 30528 df-ba 30529 df-sm 30530 df-0v 30531 df-vs 30532 df-nmcv 30533 df-ims 30534 df-ssp 30655 df-ph 30746 df-cbn 30796 df-hnorm 30901 df-hba 30902 df-hvsub 30904 df-hlim 30905 df-hcau 30906 df-sh 31140 df-ch 31154 df-oc 31185 df-ch0 31186 df-shs 31241 df-pjh 31328 |
This theorem is referenced by: axpjcl 31333 pjspansn 31510 pjimai 32109 |
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