3cubeslem1 - Mathbox for Igor Ieskov

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Theorem 3cubeslem1 43116

Description: Lemma for 3cubes 43122. (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 12903	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		0red 11147	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
5	3, 4	lttri4d 11287	. . 3 ⊢ (𝜑 → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
6		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
7		0red 11147	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ∈ ℝ)
8		peano2re 11319	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (𝐴 + 1) ∈ ℝ)
10	9	resqcld 14087	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((𝐴 + 1)↑2) ∈ ℝ)
11		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 < 0)
12	9	sqge0d 14099	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≤ ((𝐴 + 1)↑2))
13	6, 7, 10, 11, 12	ltletrd 11306	. . . . . 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 696	. . . 4 ⊢ (𝜑 → (𝐴 < 0 → 𝐴 < ((𝐴 + 1)↑2)))
16		0lt1 11672	. . . . . . . 8 ⊢ 0 < 1
17	16	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → 0 < 1)
18		id 22	. . . . . . 7 ⊢ (𝐴 = 0 → 𝐴 = 0)
19		sq1 14157	. . . . . . . 8 ⊢ (1↑2) = 1
20	19	a1i 11	. . . . . . 7 ⊢ (𝐴 = 0 → (1↑2) = 1)
21	17, 18, 20	3brtr4d 5117	. . . . . 6 ⊢ (𝐴 = 0 → 𝐴 < (1↑2))
22		0cnd 11137	. . . . . . . . 9 ⊢ (𝐴 = 0 → 0 ∈ ℂ)
23		1cnd 11139	. . . . . . . . 9 ⊢ (𝐴 = 0 → 1 ∈ ℂ)
24	18	oveq1d 7382	. . . . . . . . 9 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
25	22, 23, 24	comraddd 11360	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 + 1) = (1 + 0))
26		1p0e1 12300	. . . . . . . 8 ⊢ (1 + 0) = 1
27	25, 26	eqtrdi 2787	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 + 1) = 1)
28	27	oveq1d 7382	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 + 1)↑2) = (1↑2))
29	21, 28	breqtrrd 5113	. . . . 5 ⊢ (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2))
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 = 0 → 𝐴 < ((𝐴 + 1)↑2)))
31		ax-1rid 11108	. . . . . . . . 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 11145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 1 ∈ ℝ)
35	33, 34	readdcld 11174	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℝ)
36		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < 𝐴)
37		0red 11147	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ∈ ℝ)
38		ltle 11234	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
39	37, 33, 38	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 < 𝐴 → 0 ≤ 𝐴))
40	33	ltp1d 12086	. . . . . . . . . . 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 11673	. . . . . . . . . . 11 ⊢ 0 ≤ 1
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 ≤ 1)
46		1e0p1 12686	. . . . . . . . . . 11 ⊢ 1 = (0 + 1)
47	37, 33, 34, 36	ltadd1dd 11761	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (0 + 1) < (𝐴 + 1))
48	46, 47	eqbrtrid 5120	. . . . . . . . . 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 12011	. . . . . . . . 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 841	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) < ((𝐴 + 1) · (𝐴 + 1)))
52	32, 51	eqbrtrrd 5109	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 < ((𝐴 + 1) · (𝐴 + 1)))
53	35	recnd 11173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 + 1) ∈ ℂ)
54	53	sqvald 14105	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)))
55	52, 54	breqtrrd 5113	. . . . . 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 696	. . . 4 ⊢ (𝜑 → (0 < 𝐴 → 𝐴 < ((𝐴 + 1)↑2)))
58	15, 30, 57	3jaod 1432	. . 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 14087	. . 3 ⊢ (𝜑 → ((𝐴 + 1)↑2) ∈ ℝ)
62	3, 61	posdifd 11737	. 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 1086 = wceq 1542 ∈ wcel 2114 class class class wbr 5085 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 2c2 12236 ℚcq 12898 ↑cexp 14023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-seq 13964 df-exp 14024

This theorem is referenced by: 3cubeslem2 43117

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