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Mirrors > Home > HSE Home > Th. List > lnopeq | Structured version Visualization version GIF version |
Description: Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopeq | ⊢ ((𝑇 ∈ LinOp ∧ 𝑈 ∈ LinOp) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6351 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → (𝑇‘𝑥) = (if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥)) | |
2 | 1 | oveq1d 6828 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → ((𝑇‘𝑥) ·ih 𝑥) = ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥)) |
3 | 2 | eqeq1d 2762 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → (((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥))) |
4 | 3 | ralbidv 3124 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥))) |
5 | eqeq1 2764 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → (𝑇 = 𝑈 ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = 𝑈)) | |
6 | 4, 5 | bibi12d 334 | . 2 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, 0hop ) → ((∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈) ↔ (∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = 𝑈))) |
7 | fveq1 6351 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → (𝑈‘𝑥) = (if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥)) | |
8 | 7 | oveq1d 6828 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → ((𝑈‘𝑥) ·ih 𝑥) = ((if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥) ·ih 𝑥)) |
9 | 8 | eqeq2d 2770 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → (((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥) ·ih 𝑥))) |
10 | 9 | ralbidv 3124 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → (∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥) ·ih 𝑥))) |
11 | eqeq2 2771 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → (if(𝑇 ∈ LinOp, 𝑇, 0hop ) = 𝑈 ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = if(𝑈 ∈ LinOp, 𝑈, 0hop ))) | |
12 | 10, 11 | bibi12d 334 | . 2 ⊢ (𝑈 = if(𝑈 ∈ LinOp, 𝑈, 0hop ) → ((∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = 𝑈) ↔ (∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥) ·ih 𝑥) ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = if(𝑈 ∈ LinOp, 𝑈, 0hop )))) |
13 | 0lnop 29152 | . . . 4 ⊢ 0hop ∈ LinOp | |
14 | 13 | elimel 4294 | . . 3 ⊢ if(𝑇 ∈ LinOp, 𝑇, 0hop ) ∈ LinOp |
15 | 13 | elimel 4294 | . . 3 ⊢ if(𝑈 ∈ LinOp, 𝑈, 0hop ) ∈ LinOp |
16 | 14, 15 | lnopeqi 29176 | . 2 ⊢ (∀𝑥 ∈ ℋ ((if(𝑇 ∈ LinOp, 𝑇, 0hop )‘𝑥) ·ih 𝑥) = ((if(𝑈 ∈ LinOp, 𝑈, 0hop )‘𝑥) ·ih 𝑥) ↔ if(𝑇 ∈ LinOp, 𝑇, 0hop ) = if(𝑈 ∈ LinOp, 𝑈, 0hop )) |
17 | 6, 12, 16 | dedth2h 4284 | 1 ⊢ ((𝑇 ∈ LinOp ∧ 𝑈 ∈ LinOp) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ifcif 4230 ‘cfv 6049 (class class class)co 6813 ℋchil 28085 ·ih csp 28088 0hop ch0o 28109 LinOpclo 28113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cc 9449 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 ax-hilex 28165 ax-hfvadd 28166 ax-hvcom 28167 ax-hvass 28168 ax-hv0cl 28169 ax-hvaddid 28170 ax-hfvmul 28171 ax-hvmulid 28172 ax-hvmulass 28173 ax-hvdistr1 28174 ax-hvdistr2 28175 ax-hvmul0 28176 ax-hfi 28245 ax-his1 28248 ax-his2 28249 ax-his3 28250 ax-his4 28251 ax-hcompl 28368 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-rlim 14419 df-sum 14616 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-cn 21233 df-cnp 21234 df-lm 21235 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cfil 23253 df-cau 23254 df-cmet 23255 df-grpo 27656 df-gid 27657 df-ginv 27658 df-gdiv 27659 df-ablo 27708 df-vc 27723 df-nv 27756 df-va 27759 df-ba 27760 df-sm 27761 df-0v 27762 df-vs 27763 df-nmcv 27764 df-ims 27765 df-dip 27865 df-ssp 27886 df-ph 27977 df-cbn 28028 df-hnorm 28134 df-hba 28135 df-hvsub 28137 df-hlim 28138 df-hcau 28139 df-sh 28373 df-ch 28387 df-oc 28418 df-ch0 28419 df-shs 28476 df-pjh 28563 df-hosum 28898 df-homul 28899 df-hodif 28900 df-h0op 28916 df-lnop 29009 df-hmop 29012 |
This theorem is referenced by: leoptri 29304 |
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