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Theorem hosubsub 31683

Description: Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hosubsub	⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −_op (𝑇 −_op 𝑈)) = ((𝑆 −_op 𝑇) +_op 𝑈))

**Proof of Theorem hosubsub**
Step	Hyp	Ref	Expression
1		hosubsub2 31678	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −_op (𝑇 −_op 𝑈)) = (𝑆 +_op (𝑈 −_op 𝑇)))
2		hoaddsubass 31681	. . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑆 +_op 𝑈) −_op 𝑇) = (𝑆 +_op (𝑈 −_op 𝑇)))
3		hoaddsub 31682	. . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑆 +_op 𝑈) −_op 𝑇) = ((𝑆 −_op 𝑇) +_op 𝑈))
4	2, 3	eqtr3d 2767	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +_op (𝑈 −_op 𝑇)) = ((𝑆 −_op 𝑇) +_op 𝑈))
5	4	3com23 1123	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 +_op (𝑈 −_op 𝑇)) = ((𝑆 −_op 𝑇) +_op 𝑈))
6	1, 5	eqtrd 2765	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −_op (𝑇 −_op 𝑈)) = ((𝑆 −_op 𝑇) +_op 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ⟶wf 6543 (class class class)co 7417 ℋchba 30785 +_op chos 30804 −_op chod 30806

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30865 ax-hfvadd 30866 ax-hvcom 30867 ax-hvass 30868 ax-hv0cl 30869 ax-hvaddid 30870 ax-hfvmul 30871 ax-hvmulid 30872 ax-hvmulass 30873 ax-hvdistr1 30874 ax-hvdistr2 30875 ax-hvmul0 30876 ax-hfi 30945 ax-his1 30948 ax-his2 30949 ax-his3 30950 ax-his4 30951 ax-hcompl 31068

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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22826 df-topon 22843 df-topsp 22865 df-bases 22879 df-cld 22953 df-ntr 22954 df-cls 22955 df-nei 23032 df-cn 23161 df-cnp 23162 df-lm 23163 df-haus 23249 df-tx 23496 df-hmeo 23689 df-fil 23780 df-fm 23872 df-flim 23873 df-flf 23874 df-xms 24256 df-ms 24257 df-tms 24258 df-cfil 25213 df-cau 25214 df-cmet 25215 df-grpo 30359 df-gid 30360 df-ginv 30361 df-gdiv 30362 df-ablo 30411 df-vc 30425 df-nv 30458 df-va 30461 df-ba 30462 df-sm 30463 df-0v 30464 df-vs 30465 df-nmcv 30466 df-ims 30467 df-dip 30567 df-ssp 30588 df-ph 30679 df-cbn 30729 df-hnorm 30834 df-hba 30835 df-hvsub 30837 df-hlim 30838 df-hcau 30839 df-sh 31073 df-ch 31087 df-oc 31118 df-ch0 31119 df-shs 31174 df-pjh 31261 df-hosum 31596 df-homul 31597 df-hodif 31598 df-h0op 31614

This theorem is referenced by: hosubsub4 31684

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