opsqrlem4 - Hilbert Space Explorer

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Theorem opsqrlem4 32115

Description: Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
opsqrlem2.1	⊢ 𝑇 ∈ HrmOp
opsqrlem2.2	⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +_op ((1 / 2) ·_op (𝑇 −_op (𝑥 ∘ 𝑥)))))
opsqrlem2.3	⊢ 𝐹 = seq1(𝑆, (ℕ × { 0_hop }))

**Assertion**
Ref	Expression
opsqrlem4	⊢ 𝐹:ℕ⟶HrmOp

Distinct variable group: 𝑥,𝑦,𝑇 Allowed substitution hints: 𝑆(𝑥,𝑦) 𝐹(𝑥,𝑦)

Proof of Theorem opsqrlem4 Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12770	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12498	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		0hmop 31955	. . . . . . . 8 ⊢ 0_hop ∈ HrmOp
4	3	elexi 3459	. . . . . . 7 ⊢ 0_hop ∈ V
5	4	fvconst2 7133	. . . . . 6 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) = 0_hop )
6	5, 3	eqeltrdi 2839	. . . . 5 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) ∈ HrmOp)
7	6	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ ℕ) → ((ℕ × { 0_hop })‘𝑧) ∈ HrmOp)
8		opsqrlem2.1	. . . . . . 7 ⊢ 𝑇 ∈ HrmOp
9		opsqrlem2.2	. . . . . . 7 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +_op ((1 / 2) ·_op (𝑇 −_op (𝑥 ∘ 𝑥)))))
10		opsqrlem2.3	. . . . . . 7 ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0_hop }))
11	8, 9, 10	opsqrlem3 32114	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) = (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))))
12		halfre 12329	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
13		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → 𝑧 ∈ HrmOp)
14		eqidd 2732	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧))
15		hmopco 31995	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑧 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧)) → (𝑧 ∘ 𝑧) ∈ HrmOp)
16	13, 13, 14, 15	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) ∈ HrmOp)
17		hmopd 31994	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) ∈ HrmOp) → (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp)
18	8, 16, 17	sylancr 587	. . . . . . . 8 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp)
19		hmopm 31993	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp) → ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧))) ∈ HrmOp)
20	12, 18, 19	sylancr 587	. . . . . . 7 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧))) ∈ HrmOp)
21		hmops 31992	. . . . . . 7 ⊢ ((𝑧 ∈ HrmOp ∧ ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧))) ∈ HrmOp) → (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))) ∈ HrmOp)
22	20, 21	syldan 591	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))) ∈ HrmOp)
23	11, 22	eqeltrd 2831	. . . . 5 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) ∈ HrmOp)
24	23	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp)) → (𝑧𝑆𝑤) ∈ HrmOp)
25	1, 2, 7, 24	seqf 13925	. . 3 ⊢ (⊤ → seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
26	25	mptru 1548	. 2 ⊢ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp
27	10	feq1i 6637	. 2 ⊢ (𝐹:ℕ⟶HrmOp ↔ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
28	26, 27	mpbir 231	1 ⊢ 𝐹:ℕ⟶HrmOp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 {csn 4571 × cxp 5609 ∘ ccom 5615 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ∈ cmpo 7343 ℝcr 11000 1c1 11002 / cdiv 11769 ℕcn 12120 2c2 12175 seqcseq 13903 +_op chos 30910 ·_op chot 30911 −_op chod 30912 0_hop ch0o 30915 HrmOpcho 30922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cc 10321 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080 ax-mulf 11081 ax-hilex 30971 ax-hfvadd 30972 ax-hvcom 30973 ax-hvass 30974 ax-hv0cl 30975 ax-hvaddid 30976 ax-hfvmul 30977 ax-hvmulid 30978 ax-hvmulass 30979 ax-hvdistr1 30980 ax-hvdistr2 30981 ax-hvmul0 30982 ax-hfi 31051 ax-his1 31054 ax-his2 31055 ax-his3 31056 ax-his4 31057 ax-hcompl 31174

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-cn 23137 df-cnp 23138 df-lm 23139 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cfil 25177 df-cau 25178 df-cmet 25179 df-grpo 30465 df-gid 30466 df-ginv 30467 df-gdiv 30468 df-ablo 30517 df-vc 30531 df-nv 30564 df-va 30567 df-ba 30568 df-sm 30569 df-0v 30570 df-vs 30571 df-nmcv 30572 df-ims 30573 df-dip 30673 df-ssp 30694 df-ph 30785 df-cbn 30835 df-hnorm 30940 df-hba 30941 df-hvsub 30943 df-hlim 30944 df-hcau 30945 df-sh 31179 df-ch 31193 df-oc 31224 df-ch0 31225 df-shs 31280 df-pjh 31367 df-hosum 31702 df-homul 31703 df-hodif 31704 df-h0op 31720 df-hmop 31816

This theorem is referenced by: opsqrlem5 32116 opsqrlem6 32117

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