3cubeslem1 - Mathbox for Igor Ieskov

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Theorem 3cubeslem1 42700

Description: Lemma for 3cubes 42706. (Contributed by Igor Ieskov, 22-Jan-2024.)

**Hypothesis**
Ref	Expression
3cubeslem1.a	⊢ (𝜑 → 𝐴 ∈ ℚ)

**Assertion**
Ref	Expression
3cubeslem1	⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))

**Proof of Theorem 3cubeslem1**
Step	Hyp	Ref	Expression
1		3cubeslem1.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℚ)
2		qre 12996	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		0red 11265	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
5	3, 4	lttri4d 11403	. . 3 ⊢ (𝜑 → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
6		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
7		0red 11265	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ∈ ℝ)
8		peano2re 11435	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (𝐴 + 1) ∈ ℝ)
10	9	resqcld 14166	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((𝐴 + 1)↑2) ∈ ℝ)
11		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < 0)
12	9	sqge0d 14178	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≤ ((𝐴 + 1)↑2))
13	6, 7, 10, 11, 12	ltletrd 11422	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < ((𝐴 + 1)↑2))
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < ((𝐴 + 1)↑2)))
15	3, 14	mpand 695	. . . 4 ⊢ (𝜑 → (𝐴 < 0 → 𝐴 < ((𝐴 + 1)↑2)))
16		0lt1 11786	. . . . . . . 8 ⊢ 0 < 1
17	16	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → 0 < 1)
18		id 22	. . . . . . 7 ⊢ (𝐴 = 0 → 𝐴 = 0)
19		sq1 14235	. . . . . . . 8 ⊢ (1↑2) = 1
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → (1↑2) = 1)
21	17, 18, 20	3brtr4d 5174	. . . . . 6 ⊢ (𝐴 = 0 → 𝐴 < (1↑2))
22		0cnd 11255	. . . . . . . . 9 ⊢ (𝐴 = 0 → 0 ∈ ℂ)
23		1cnd 11257	. . . . . . . . 9 ⊢ (𝐴 = 0 → 1 ∈ ℂ)
24	18	oveq1d 7447	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
25	22, 23, 24	comraddd 11476	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 1) = (1 + 0))
26		1p0e1 12391	. . . . . . . 8 ⊢ (1 + 0) = 1
27	25, 26	eqtrdi 2792	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 + 1) = 1)
28	27	oveq1d 7447	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 + 1)↑2) = (1↑2))
29	21, 28	breqtrrd 5170	. . . . 5 ⊢ (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2))
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2)))
31		ax-1rid 11226	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) = 𝐴)
33		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
34		1red 11263	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 ∈ ℝ)
35	33, 34	readdcld 11291	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℝ)
36		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < 𝐴)
37		0red 11265	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ∈ ℝ)
38		ltle 11350	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
39	37, 33, 38	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 < 𝐴 → 0 ≤ 𝐴))
40	33	ltp1d 12199	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < (𝐴 + 1))
41	39, 40	jctird 526	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 < 𝐴 → (0 ≤ 𝐴 ∧ 𝐴 < (𝐴 + 1))))
42	36, 41	mpd 15	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 ≤ 𝐴 ∧ 𝐴 < (𝐴 + 1)))
43	34, 35	jca 511	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ))
44		0le1 11787	. . . . . . . . . . 11 ⊢ 0 ≤ 1
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 1)
46		1e0p1 12777	. . . . . . . . . . 11 ⊢ 1 = (0 + 1)
47	37, 33, 34, 36	ltadd1dd 11875	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 + 1) < (𝐴 + 1))
48	46, 47	eqbrtrid 5177	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 < (𝐴 + 1))
49	43, 45, 48	jca32 515	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ (0 ≤ 1 ∧ 1 < (𝐴 + 1))))
50		ltmul12a 12124	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < (𝐴 + 1))) ∧ ((1 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ (0 ≤ 1 ∧ 1 < (𝐴 + 1)))) → (𝐴 · 1) < ((𝐴 + 1) · (𝐴 + 1)))
51	33, 35, 42, 49, 50	syl1111anc 840	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) < ((𝐴 + 1) · (𝐴 + 1)))
52	32, 51	eqbrtrrd 5166	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1) · (𝐴 + 1)))
53	35	recnd 11290	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℂ)
54	53	sqvald 14184	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)))
55	52, 54	breqtrrd 5170	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1)↑2))
56	55	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1)↑2)))
57	3, 56	mpand 695	. . . 4 ⊢ (𝜑 → (0 < 𝐴 → 𝐴 < ((𝐴 + 1)↑2)))
58	15, 30, 57	3jaod 1430	. . 3 ⊢ (𝜑 → ((𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴) → 𝐴 < ((𝐴 + 1)↑2)))
59	5, 58	mpd 15	. 2 ⊢ (𝜑 → 𝐴 < ((𝐴 + 1)↑2))
60	3, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ)
61	60	resqcld 14166	. . 3 ⊢ (𝜑 → ((𝐴 + 1)↑2) ∈ ℝ)
62	3, 61	posdifd 11851	. 2 ⊢ (𝜑 → (𝐴 < ((𝐴 + 1)↑2) ↔ 0 < (((𝐴 + 1)↑2) − 𝐴)))
63	59, 62	mpbid 232	1 ⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1085 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 < clt 11296 ≤ cle 11297 − cmin 11493 2c2 12322 ℚcq 12991 ↑cexp 14103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-seq 14044 df-exp 14104

This theorem is referenced by: 3cubeslem2 42701

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