opsqrlem4 - Hilbert Space Explorer

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Theorem opsqrlem4 32079

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 12843	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12571	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		0hmop 31919	. . . . . . . 8 ⊢ 0_hop ∈ HrmOp
4	3	elexi 3473	. . . . . . 7 ⊢ 0_hop ∈ V
5	4	fvconst2 7181	. . . . . 6 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0_hop })‘𝑧) = 0_hop )
6	5, 3	eqeltrdi 2837	. . . . 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 32078	. . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) = (𝑧 +_op ((1 / 2) ·_op (𝑇 −_op (𝑧 ∘ 𝑧)))))
12		halfre 12402	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
13		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → 𝑧 ∈ HrmOp)
14		eqidd 2731	. . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧))
15		hmopco 31959	. . . . . . . . . 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 31958	. . . . . . . . 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 31957	. . . . . . . 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 31956	. . . . . . 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 2829	. . . . 5 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) ∈ HrmOp)
24	23	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp)) → (𝑧𝑆𝑤) ∈ HrmOp)
25	1, 2, 7, 24	seqf 13995	. . 3 ⊢ (⊤ → seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp)
26	25	mptru 1547	. 2 ⊢ seq1(𝑆, (ℕ × { 0_hop })):ℕ⟶HrmOp
27	10	feq1i 6682	. 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 4592 × cxp 5639 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ℝcr 11074 1c1 11076 / cdiv 11842 ℕcn 12193 2c2 12248 seqcseq 13973 +_op chos 30874 ·_op chot 30875 −_op chod 30876 0_hop ch0o 30879 HrmOpcho 30886

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 ax-hilex 30935 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hv0cl 30939 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvmulass 30943 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 ax-hfi 31015 ax-his1 31018 ax-his2 31019 ax-his3 31020 ax-his4 31021 ax-hcompl 31138

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-cn 23121 df-cnp 23122 df-lm 23123 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cfil 25162 df-cau 25163 df-cmet 25164 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-dip 30637 df-ssp 30658 df-ph 30749 df-cbn 30799 df-hnorm 30904 df-hba 30905 df-hvsub 30907 df-hlim 30908 df-hcau 30909 df-sh 31143 df-ch 31157 df-oc 31188 df-ch0 31189 df-shs 31244 df-pjh 31331 df-hosum 31666 df-homul 31667 df-hodif 31668 df-h0op 31684 df-hmop 31780

This theorem is referenced by: opsqrlem5 32080 opsqrlem6 32081

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