3cubeslem1 - Mathbox for Igor Ieskov

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Theorem 3cubeslem1 40506

Description: Lemma for 3cubes 40512. (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 12693	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		0red 10978	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
5	3, 4	lttri4d 11116	. . 3 ⊢ (𝜑 → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
6		simpl 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
7		0red 10978	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ∈ ℝ)
8		peano2re 11148	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
9	8	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (𝐴 + 1) ∈ ℝ)
10	9	resqcld 13965	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((𝐴 + 1)↑2) ∈ ℝ)
11		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < 0)
12	9	sqge0d 13966	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≤ ((𝐴 + 1)↑2))
13	6, 7, 10, 11, 12	ltletrd 11135	. . . . . 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 692	. . . 4 ⊢ (𝜑 → (𝐴 < 0 → 𝐴 < ((𝐴 + 1)↑2)))
16		0lt1 11497	. . . . . . . 8 ⊢ 0 < 1
17	16	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → 0 < 1)
18		id 22	. . . . . . 7 ⊢ (𝐴 = 0 → 𝐴 = 0)
19		sq1 13912	. . . . . . . 8 ⊢ (1↑2) = 1
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → (1↑2) = 1)
21	17, 18, 20	3brtr4d 5106	. . . . . 6 ⊢ (𝐴 = 0 → 𝐴 < (1↑2))
22		0cnd 10968	. . . . . . . . 9 ⊢ (𝐴 = 0 → 0 ∈ ℂ)
23		1cnd 10970	. . . . . . . . 9 ⊢ (𝐴 = 0 → 1 ∈ ℂ)
24	18	oveq1d 7290	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
25	22, 23, 24	comraddd 11189	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 1) = (1 + 0))
26		1p0e1 12097	. . . . . . . 8 ⊢ (1 + 0) = 1
27	25, 26	eqtrdi 2794	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 + 1) = 1)
28	27	oveq1d 7290	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 + 1)↑2) = (1↑2))
29	21, 28	breqtrrd 5102	. . . . 5 ⊢ (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2))
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2)))
31		ax-1rid 10941	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
32	31	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) = 𝐴)
33		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
34		1red 10976	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 ∈ ℝ)
35	33, 34	readdcld 11004	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℝ)
36		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < 𝐴)
37		0red 10978	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ∈ ℝ)
38		ltle 11063	. . . . . . . . . . . 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 11905	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < (𝐴 + 1))
41	39, 40	jctird 527	. . . . . . . . . 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 512	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ))
44		0le1 11498	. . . . . . . . . . 11 ⊢ 0 ≤ 1
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 1)
46		1e0p1 12479	. . . . . . . . . . 11 ⊢ 1 = (0 + 1)
47	37, 33, 34, 36	ltadd1dd 11586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 + 1) < (𝐴 + 1))
48	46, 47	eqbrtrid 5109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 < (𝐴 + 1))
49	43, 45, 48	jca32 516	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ (0 ≤ 1 ∧ 1 < (𝐴 + 1))))
50		ltmul12a 11831	. . . . . . . . 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 837	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) < ((𝐴 + 1) · (𝐴 + 1)))
52	32, 51	eqbrtrrd 5098	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1) · (𝐴 + 1)))
53	35	recnd 11003	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℂ)
54	53	sqvald 13861	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)))
55	52, 54	breqtrrd 5102	. . . . . 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 692	. . . 4 ⊢ (𝜑 → (0 < 𝐴 → 𝐴 < ((𝐴 + 1)↑2)))
58	15, 30, 57	3jaod 1427	. . 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 13965	. . 3 ⊢ (𝜑 → ((𝐴 + 1)↑2) ∈ ℝ)
62	3, 61	posdifd 11562	. 2 ⊢ (𝜑 → (𝐴 < ((𝐴 + 1)↑2) ↔ 0 < (((𝐴 + 1)↑2) − 𝐴)))
63	59, 62	mpbid 231	1 ⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 2c2 12028 ℚcq 12688 ↑cexp 13782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-seq 13722 df-exp 13783

This theorem is referenced by: 3cubeslem2 40507

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