opsqrlem4 - Hilbert Space Explorer

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Theorem opsqrlem4 32045

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 12812	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12540	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		0hmop 31885	. . . . . . . 8 ⊢ 0_hop ∈ HrmOp
4	3	elexi 3467	. . . . . . 7 ⊢ 0_hop ∈ V
5	4	fvconst2 7160	. . . . . 6 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) = 0_hop )
6	5, 3	eqeltrdi 2836	. . . . 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 32044	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) = (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))))
12		halfre 12371	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
13		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → 𝑧 ∈ HrmOp)
14		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧))
15		hmopco 31925	. . . . . . . . . 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 31924	. . . . . . . . 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 31923	. . . . . . . 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 31922	. . . . . . 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 2828	. . . . 5 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) ∈ HrmOp)
24	23	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp)) → (𝑧𝑆𝑤) ∈ HrmOp)
25	1, 2, 7, 24	seqf 13964	. . 3 ⊢ (⊤ → seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
26	25	mptru 1547	. 2 ⊢ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp
27	10	feq1i 6661	. 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 1540 ⊤wtru 1541 ∈ wcel 2109 {csn 4585 × cxp 5629 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ℝcr 11043 1c1 11045 / cdiv 11811 ℕcn 12162 2c2 12217 seqcseq 13942 +_op chos 30840 ·_op chot 30841 −_op chod 30842 0_hop ch0o 30845 HrmOpcho 30852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 ax-hcompl 31104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-cn 23090 df-cnp 23091 df-lm 23092 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cfil 25131 df-cau 25132 df-cmet 25133 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-dip 30603 df-ssp 30624 df-ph 30715 df-cbn 30765 df-hnorm 30870 df-hba 30871 df-hvsub 30873 df-hlim 30874 df-hcau 30875 df-sh 31109 df-ch 31123 df-oc 31154 df-ch0 31155 df-shs 31210 df-pjh 31297 df-hosum 31632 df-homul 31633 df-hodif 31634 df-h0op 31650 df-hmop 31746

This theorem is referenced by: opsqrlem5 32046 opsqrlem6 32047

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