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Theorem resqrexlemsqa 11053

Description: Lemma for resqrex 11055. The square of a limit is 𝐴. (Contributed by Jim Kingdon, 7-Aug-2021.)

**Hypotheses**
Ref	Expression
resqrexlemex.seq	⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))
resqrexlemex.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
resqrexlemex.agt0	⊢ (𝜑 → 0 ≤ 𝐴)
resqrexlemgt0.rr	⊢ (𝜑 → 𝐿 ∈ ℝ)
resqrexlemgt0.lim	⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))

**Assertion**
Ref	Expression
resqrexlemsqa	⊢ (𝜑 → (𝐿↑2) = 𝐴)

Distinct variable groups: 𝐴,𝑒,𝑗 𝑦,𝐴,𝑧 𝑒,𝐹,𝑗 𝑦,𝐹,𝑧 𝑖,𝐹 𝑒,𝐿,𝑗,𝑖 𝑦,𝐿,𝑧 𝑒,𝑖,𝑗 𝜑,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑒,𝑖,𝑗) 𝐴(𝑖)

Proof of Theorem resqrexlemsqa Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resqrexlemex.seq	. . . . . . 7 ⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))
2		resqrexlemex.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		resqrexlemex.agt0	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴)
4	1, 2, 3	resqrexlemf 11036	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5	4	ffvelcdmda 5668	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ⁺)
6		2z 9301	. . . . . 6 ⊢ 2 ∈ ℤ
7	6	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 2 ∈ ℤ)
8	5, 7	rpexpcld 10698	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥)↑2) ∈ ℝ⁺)
9		eqid 2189	. . . 4 ⊢ (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))
10	8, 9	fmptd 5687	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)):ℕ⟶ℝ⁺)
11		rpssre 9684	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
13	10, 12	fssd 5394	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)):ℕ⟶ℝ)
14		resqrexlemgt0.rr	. . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ)
15	14	resqcld 10700	. 2 ⊢ (𝜑 → (𝐿↑2) ∈ ℝ)
16		resqrexlemgt0.lim	. . . 4 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))
17		oveq2 5900	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → (𝐿 + 𝑒) = (𝐿 + 𝑎))
18	17	breq2d 4030	. . . . . . . 8 ⊢ (𝑒 = 𝑎 → ((𝐹‘𝑖) < (𝐿 + 𝑒) ↔ (𝐹‘𝑖) < (𝐿 + 𝑎)))
19		oveq2 5900	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → ((𝐹‘𝑖) + 𝑒) = ((𝐹‘𝑖) + 𝑎))
20	19	breq2d 4030	. . . . . . . 8 ⊢ (𝑒 = 𝑎 → (𝐿 < ((𝐹‘𝑖) + 𝑒) ↔ 𝐿 < ((𝐹‘𝑖) + 𝑎)))
21	18, 20	anbi12d 473	. . . . . . 7 ⊢ (𝑒 = 𝑎 → (((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)) ↔ ((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎))))
22	21	rexralbidv 2516	. . . . . 6 ⊢ (𝑒 = 𝑎 → (∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)) ↔ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎))))
23	22	cbvralv 2718	. . . . 5 ⊢ (∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)) ↔ ∀𝑎 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)))
24		fveq2 5531	. . . . . . . 8 ⊢ (𝑗 = 𝑏 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑏))
25	24	raleqdv 2692	. . . . . . 7 ⊢ (𝑗 = 𝑏 → (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎))))
26	25	cbvrexv 2719	. . . . . 6 ⊢ (∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∃𝑏 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)))
27	26	ralbii 2496	. . . . 5 ⊢ (∀𝑎 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)))
28		fveq2 5531	. . . . . . . . . 10 ⊢ (𝑖 = 𝑐 → (𝐹‘𝑖) = (𝐹‘𝑐))
29	28	breq1d 4028	. . . . . . . . 9 ⊢ (𝑖 = 𝑐 → ((𝐹‘𝑖) < (𝐿 + 𝑎) ↔ (𝐹‘𝑐) < (𝐿 + 𝑎)))
30	28	oveq1d 5907	. . . . . . . . . 10 ⊢ (𝑖 = 𝑐 → ((𝐹‘𝑖) + 𝑎) = ((𝐹‘𝑐) + 𝑎))
31	30	breq2d 4030	. . . . . . . . 9 ⊢ (𝑖 = 𝑐 → (𝐿 < ((𝐹‘𝑖) + 𝑎) ↔ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
32	29, 31	anbi12d 473	. . . . . . . 8 ⊢ (𝑖 = 𝑐 → (((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎))))
33	32	cbvralv 2718	. . . . . . 7 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∀𝑐 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
34	33	rexbii 2497	. . . . . 6 ⊢ (∃𝑏 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∃𝑏 ∈ ℕ ∀𝑐 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
35	34	ralbii 2496	. . . . 5 ⊢ (∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑖) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑎)) ↔ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑐 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
36	23, 27, 35	3bitri 206	. . . 4 ⊢ (∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)) ↔ ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑐 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
37	16, 36	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑐 ∈ (ℤ_≥‘𝑏)((𝐹‘𝑐) < (𝐿 + 𝑎) ∧ 𝐿 < ((𝐹‘𝑐) + 𝑎)))
38	1, 2, 3, 14, 37, 9	resqrexlemglsq 11051	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑑 ∈ (ℤ_≥‘𝑏)(((𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))‘𝑑) < ((𝐿↑2) + 𝑎) ∧ (𝐿↑2) < (((𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))‘𝑑) + 𝑎)))
39	1, 2, 3, 14, 37, 9	resqrexlemga 11052	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℕ ∀𝑑 ∈ (ℤ_≥‘𝑏)(((𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))‘𝑑) < (𝐴 + 𝑎) ∧ 𝐴 < (((𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))‘𝑑) + 𝑎)))
40	13, 15, 38, 2, 39	recvguniq 11024	1 ⊢ (𝜑 → (𝐿↑2) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 ⊆ wss 3144 {csn 3607 class class class wbr 4018 ↦ cmpt 4079 × cxp 4639 ‘cfv 5232 (class class class)co 5892 ∈ cmpo 5894 ℝcr 7830 0cc0 7831 1c1 7832 + caddc 7834 < clt 8012 ≤ cle 8013 / cdiv 8649 ℕcn 8939 2c2 8990 ℤcz 9273 ℤ_≥cuz 9548 ℝ⁺crp 9673 seqcseq 10465 ↑cexp 10539

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-mulrcl 7930 ax-addcom 7931 ax-mulcom 7932 ax-addass 7933 ax-mulass 7934 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-1rid 7938 ax-0id 7939 ax-rnegex 7940 ax-precex 7941 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-apti 7946 ax-pre-ltadd 7947 ax-pre-mulgt0 7948 ax-pre-mulext 7949 ax-arch 7950

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-reap 8552 df-ap 8559 df-div 8650 df-inn 8940 df-2 8998 df-3 8999 df-4 9000 df-n0 9197 df-z 9274 df-uz 9549 df-rp 9674 df-seqfrec 10466 df-exp 10540

This theorem is referenced by: resqrexlemex 11054

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