dp2lt - Mathbox for Thierry Arnoux

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Theorem dp2lt 32653

Description: Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.)

**Hypotheses**
Ref	Expression
dp2lt.a	⊢ 𝐴 ∈ ℕ₀
dp2lt.b	⊢ 𝐵 ∈ ℝ⁺
dp2lt.c	⊢ 𝐶 ∈ ℝ⁺
dp2lt.l	⊢ 𝐵 < 𝐶

**Assertion**
Ref	Expression
dp2lt	⊢ _𝐴𝐵 < _𝐴𝐶

**Proof of Theorem dp2lt**
Step	Hyp	Ref	Expression
1		rpssre 13013	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
2		dp2lt.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
3	1, 2	sselii 3969	. . . . 5 ⊢ 𝐵 ∈ ℝ
4		10re 12726	. . . . 5 ⊢ ;10 ∈ ℝ
5		0re 11246	. . . . . 6 ⊢ 0 ∈ ℝ
6		10pos 12724	. . . . . 6 ⊢ 0 < ;10
7	5, 6	gtneii 11356	. . . . 5 ⊢ ;10 ≠ 0
8		redivcl 11963	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐵 / ;10) ∈ ℝ)
9	3, 4, 7, 8	mp3an 1457	. . . 4 ⊢ (𝐵 / ;10) ∈ ℝ
10		dp2lt.c	. . . . . 6 ⊢ 𝐶 ∈ ℝ⁺
11	1, 10	sselii 3969	. . . . 5 ⊢ 𝐶 ∈ ℝ
12		redivcl 11963	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐶 / ;10) ∈ ℝ)
13	11, 4, 7, 12	mp3an 1457	. . . 4 ⊢ (𝐶 / ;10) ∈ ℝ
14		dp2lt.a	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
15	14	nn0rei 12513	. . . 4 ⊢ 𝐴 ∈ ℝ
16	9, 13, 15	3pm3.2i 1336	. . 3 ⊢ ((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ)
17		dp2lt.l	. . . 4 ⊢ 𝐵 < 𝐶
18	4, 6	pm3.2i 469	. . . . 5 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
19		ltdiv1 12108	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (;10 ∈ ℝ ∧ 0 < ;10)) → (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10)))
20	3, 11, 18, 19	mp3an 1457	. . . 4 ⊢ (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10))
21	17, 20	mpbi 229	. . 3 ⊢ (𝐵 / ;10) < (𝐶 / ;10)
22		axltadd 11317	. . . 4 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < (𝐶 / ;10) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10))))
23	22	imp 405	. . 3 ⊢ ((((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝐵 / ;10) < (𝐶 / ;10)) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)))
24	16, 21, 23	mp2an 690	. 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10))
25		df-dp2 32640	. 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
26		df-dp2 32640	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
27	24, 25, 26	3brtr4i 5173	1 ⊢ _𝐴𝐵 < _𝐴𝐶

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 (class class class)co 7416 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 < clt 11278 / cdiv 11901 ℕ₀cn0 12502 cdc 12707 ℝ⁺crp 13006 cdp2 32639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-dec 12708 df-rp 13007 df-dp2 32640

This theorem is referenced by: dplt 32672 hgt750lem2 34341

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