opsqrlem4 - Hilbert Space Explorer

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Theorem opsqrlem4 29919

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 12280	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12012	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		0hmop 29759	. . . . . . . 8 ⊢ 0_hop ∈ HrmOp
4	3	elexi 3513	. . . . . . 7 ⊢ 0_hop ∈ V
5	4	fvconst2 6965	. . . . . 6 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) = 0_hop )
6	5, 3	eqeltrdi 2921	. . . . 5 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) ∈ HrmOp)
7	6	adantl 484	. . . 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 29918	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) = (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))))
12		halfre 11850	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
13		simpl 485	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → 𝑧 ∈ HrmOp)
14		eqidd 2822	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧))
15		hmopco 29799	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑧 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧)) → (𝑧 ∘ 𝑧) ∈ HrmOp)
16	13, 13, 14, 15	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) ∈ HrmOp)
17		hmopd 29798	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) ∈ HrmOp) → (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp)
18	8, 16, 17	sylancr 589	. . . . . . . 8 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp)
19		hmopm 29797	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ (𝑇 −_op (𝑧 ∘ 𝑧)) ∈ HrmOp) → ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧))) ∈ HrmOp)
20	12, 18, 19	sylancr 589	. . . . . . 7 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧))) ∈ HrmOp)
21		hmops 29796	. . . . . . 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 593	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))) ∈ HrmOp)
23	11, 22	eqeltrd 2913	. . . . 5 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) ∈ HrmOp)
24	23	adantl 484	. . . 4 ⊢ ((⊤ ∧ (𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp)) → (𝑧𝑆𝑤) ∈ HrmOp)
25	1, 2, 7, 24	seqf 13390	. . 3 ⊢ (⊤ → seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
26	25	mptru 1540	. 2 ⊢ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp
27	10	feq1i 6504	. 2 ⊢ (𝐹:ℕ⟶HrmOp ↔ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
28	26, 27	mpbir 233	1 ⊢ 𝐹:ℕ⟶HrmOp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 {csn 4566 × cxp 5552 ∘ ccom 5558 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 ℝcr 10535 1c1 10537 / cdiv 11296 ℕcn 11637 2c2 11691 seqcseq 13368 +_op chos 28714 ·_op chot 28715 −_op chod 28716 0_hop ch0o 28719 HrmOpcho 28726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cc 9856 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 ax-hilex 28775 ax-hfvadd 28776 ax-hvcom 28777 ax-hvass 28778 ax-hv0cl 28779 ax-hvaddid 28780 ax-hfvmul 28781 ax-hvmulid 28782 ax-hvmulass 28783 ax-hvdistr1 28784 ax-hvdistr2 28785 ax-hvmul0 28786 ax-hfi 28855 ax-his1 28858 ax-his2 28859 ax-his3 28860 ax-his4 28861 ax-hcompl 28978

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-omul 8106 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-acn 9370 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-cn 21834 df-cnp 21835 df-lm 21836 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cfil 23857 df-cau 23858 df-cmet 23859 df-grpo 28269 df-gid 28270 df-ginv 28271 df-gdiv 28272 df-ablo 28321 df-vc 28335 df-nv 28368 df-va 28371 df-ba 28372 df-sm 28373 df-0v 28374 df-vs 28375 df-nmcv 28376 df-ims 28377 df-dip 28477 df-ssp 28498 df-ph 28589 df-cbn 28639 df-hnorm 28744 df-hba 28745 df-hvsub 28747 df-hlim 28748 df-hcau 28749 df-sh 28983 df-ch 28997 df-oc 29028 df-ch0 29029 df-shs 29084 df-pjh 29171 df-hosum 29506 df-homul 29507 df-hodif 29508 df-h0op 29524 df-hmop 29620

This theorem is referenced by: opsqrlem5 29920 opsqrlem6 29921

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