climlec3 - Mathbox for Scott Fenton

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Theorem climlec3 35556

Description: Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)

**Hypotheses**
Ref	Expression
climlec3.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
climlec3.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
climlec3.3	⊢ (𝜑 → 𝐵 ∈ ℝ)
climlec3.4	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climlec3.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
climlec3.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵)

**Assertion**
Ref	Expression
climlec3	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝐹 𝜑,𝑘 𝑘,𝑀 𝑘,𝑍

Proof of Theorem climlec3 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climlec3.1	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		climlec3.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climlec3.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	renegcld 11691	. . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ)
5		climlec3.4	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
6		0cnd 11257	. . . . 5 ⊢ (𝜑 → 0 ∈ ℂ)
7	1	fvexi 6915	. . . . . . 7 ⊢ 𝑍 ∈ V
8	7	mptex 7240	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V)
10		climlec3.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
11	10	recnd 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
12		eqid 2726	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) = (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))
13		fveq2 6901	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
14	13	negeqd 11504	. . . . . . 7 ⊢ (𝑚 = 𝑘 → -(𝐹‘𝑚) = -(𝐹‘𝑘))
15		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
16	10	renegcld 11691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ)
17	12, 14, 15, 16	fvmptd3 7032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = -(𝐹‘𝑘))
18		df-neg 11497	. . . . . 6 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘))
19	17, 18	eqtrdi 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = (0 − (𝐹‘𝑘)))
20	1, 2, 5, 6, 9, 11, 19	climsubc2 15641	. . . 4 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ (0 − 𝐴))
21		df-neg 11497	. . . 4 ⊢ -𝐴 = (0 − 𝐴)
22	20, 21	breqtrrdi 5195	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ -𝐴)
23	17, 16	eqeltrd 2826	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) ∈ ℝ)
24		climlec3.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵)
25	3	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
26	10, 25	lenegd 11843	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≤ 𝐵 ↔ -𝐵 ≤ -(𝐹‘𝑘)))
27	24, 26	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ -(𝐹‘𝑘))
28	27, 17	breqtrrd 5181	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘))
29	1, 2, 4, 22, 23, 28	climlec2 15663	. 2 ⊢ (𝜑 → -𝐵 ≤ -𝐴)
30	1, 2, 5, 10	climrecl 15585	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31	30, 3	lenegd 11843	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
32	29, 31	mpbird 256	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 class class class wbr 5153 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 0cc0 11158 ≤ cle 11299 − cmin 11494 -cneg 11495 ℤcz 12610 ℤ_≥cuz 12874 ⇝ cli 15486

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fl 13812 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491

This theorem is referenced by: (None)

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