3cubeslem1 - Mathbox for Igor Ieskov

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Theorem 3cubeslem1 42674

Description: Lemma for 3cubes 42680. (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 12974	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		0red 11243	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
5	3, 4	lttri4d 11381	. . 3 ⊢ (𝜑 → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
6		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
7		0red 11243	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ∈ ℝ)
8		peano2re 11413	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (𝐴 + 1) ∈ ℝ)
10	9	resqcld 14148	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((𝐴 + 1)↑2) ∈ ℝ)
11		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < 0)
12	9	sqge0d 14160	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≤ ((𝐴 + 1)↑2))
13	6, 7, 10, 11, 12	ltletrd 11400	. . . . . 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 11764	. . . . . . . 8 ⊢ 0 < 1
17	16	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → 0 < 1)
18		id 22	. . . . . . 7 ⊢ (𝐴 = 0 → 𝐴 = 0)
19		sq1 14218	. . . . . . . 8 ⊢ (1↑2) = 1
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → (1↑2) = 1)
21	17, 18, 20	3brtr4d 5156	. . . . . 6 ⊢ (𝐴 = 0 → 𝐴 < (1↑2))
22		0cnd 11233	. . . . . . . . 9 ⊢ (𝐴 = 0 → 0 ∈ ℂ)
23		1cnd 11235	. . . . . . . . 9 ⊢ (𝐴 = 0 → 1 ∈ ℂ)
24	18	oveq1d 7425	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
25	22, 23, 24	comraddd 11454	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 1) = (1 + 0))
26		1p0e1 12369	. . . . . . . 8 ⊢ (1 + 0) = 1
27	25, 26	eqtrdi 2787	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 + 1) = 1)
28	27	oveq1d 7425	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 + 1)↑2) = (1↑2))
29	21, 28	breqtrrd 5152	. . . . 5 ⊢ (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2))
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2)))
31		ax-1rid 11204	. . . . . . . . 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 11241	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 ∈ ℝ)
35	33, 34	readdcld 11269	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℝ)
36		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < 𝐴)
37		0red 11243	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ∈ ℝ)
38		ltle 11328	. . . . . . . . . . . 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 12177	. . . . . . . . . . 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 11765	. . . . . . . . . . 11 ⊢ 0 ≤ 1
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 1)
46		1e0p1 12755	. . . . . . . . . . 11 ⊢ 1 = (0 + 1)
47	37, 33, 34, 36	ltadd1dd 11853	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 + 1) < (𝐴 + 1))
48	46, 47	eqbrtrid 5159	. . . . . . . . . 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 12102	. . . . . . . . 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 5148	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1) · (𝐴 + 1)))
53	35	recnd 11268	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℂ)
54	53	sqvald 14166	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)))
55	52, 54	breqtrrd 5152	. . . . . 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 1431	. . 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 14148	. . 3 ⊢ (𝜑 → ((𝐴 + 1)↑2) ∈ ℝ)
62	3, 61	posdifd 11829	. 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 1540 ∈ wcel 2109 class class class wbr 5124 (class class class)co 7410 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 < clt 11274 ≤ cle 11275 − cmin 11471 2c2 12300 ℚcq 12969 ↑cexp 14084

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 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-seq 14025 df-exp 14085

This theorem is referenced by: 3cubeslem2 42675

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