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Theorem resqrexlemsqa 11046

Description: Lemma for resqrex 11048. 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 11029	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5	4	ffvelcdmda 5664	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ⁺)
6		2z 9294	. . . . . 6 ⊢ 2 ∈ ℤ
7	6	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 2 ∈ ℤ)
8	5, 7	rpexpcld 10691	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥)↑2) ∈ ℝ⁺)
9		eqid 2187	. . . 4 ⊢ (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))
10	8, 9	fmptd 5683	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)):ℕ⟶ℝ⁺)
11		rpssre 9677	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
13	10, 12	fssd 5390	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)):ℕ⟶ℝ)
14		resqrexlemgt0.rr	. . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ)
15	14	resqcld 10693	. 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 5896	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → (𝐿 + 𝑒) = (𝐿 + 𝑎))
18	17	breq2d 4027	. . . . . . . 8 ⊢ (𝑒 = 𝑎 → ((𝐹‘𝑖) < (𝐿 + 𝑒) ↔ (𝐹‘𝑖) < (𝐿 + 𝑎)))
19		oveq2 5896	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → ((𝐹‘𝑖) + 𝑒) = ((𝐹‘𝑖) + 𝑎))
20	19	breq2d 4027	. . . . . . . 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 2513	. . . . . 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 2715	. . . . 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 5527	. . . . . . . 8 ⊢ (𝑗 = 𝑏 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑏))
25	24	raleqdv 2689	. . . . . . 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 2716	. . . . . 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 2493	. . . . 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 5527	. . . . . . . . . 10 ⊢ (𝑖 = 𝑐 → (𝐹‘𝑖) = (𝐹‘𝑐))
29	28	breq1d 4025	. . . . . . . . 9 ⊢ (𝑖 = 𝑐 → ((𝐹‘𝑖) < (𝐿 + 𝑎) ↔ (𝐹‘𝑐) < (𝐿 + 𝑎)))
30	28	oveq1d 5903	. . . . . . . . . 10 ⊢ (𝑖 = 𝑐 → ((𝐹‘𝑖) + 𝑎) = ((𝐹‘𝑐) + 𝑎))
31	30	breq2d 4027	. . . . . . . . 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 2715	. . . . . . 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 2494	. . . . . 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 2493	. . . . 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 11044	. 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 11045	. 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 11017	1 ⊢ (𝜑 → (𝐿↑2) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 ⊆ wss 3141 {csn 3604 class class class wbr 4015 ↦ cmpt 4076 × cxp 4636 ‘cfv 5228 (class class class)co 5888 ∈ cmpo 5890 ℝcr 7823 0cc0 7824 1c1 7825 + caddc 7827 < clt 8005 ≤ cle 8006 / cdiv 8642 ℕcn 8932 2c2 8983 ℤcz 9266 ℤ_≥cuz 9541 ℝ⁺crp 9666 seqcseq 10458 ↑cexp 10532

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-rp 9667 df-seqfrec 10459 df-exp 10533

This theorem is referenced by: resqrexlemex 11047

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