Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpadd2 | Structured version Visualization version GIF version |
Description: Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
Ref | Expression |
---|---|
dpadd2.a | ⊢ 𝐴 ∈ ℕ0 |
dpadd2.b | ⊢ 𝐵 ∈ ℝ+ |
dpadd2.c | ⊢ 𝐶 ∈ ℕ0 |
dpadd2.d | ⊢ 𝐷 ∈ ℝ+ |
dpadd2.e | ⊢ 𝐸 ∈ ℕ0 |
dpadd2.f | ⊢ 𝐹 ∈ ℝ+ |
dpadd2.g | ⊢ 𝐺 ∈ ℕ0 |
dpadd2.h | ⊢ 𝐻 ∈ ℕ0 |
dpadd2.i | ⊢ (𝐺 + 𝐻) = 𝐼 |
dpadd2.1 | ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) |
Ref | Expression |
---|---|
dpadd2 | ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpadd2.g | . . . 4 ⊢ 𝐺 ∈ ℕ0 | |
2 | dpadd2.a | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
3 | 2 | nn0rei 11911 | . . . . 5 ⊢ 𝐴 ∈ ℝ |
4 | dpadd2.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
5 | rpre 12400 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
7 | dp2cl 30560 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ) | |
8 | 3, 6, 7 | mp2an 690 | . . . 4 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 1, 8 | dpval2 30573 | . . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10)) |
10 | dpadd2.h | . . . 4 ⊢ 𝐻 ∈ ℕ0 | |
11 | dpadd2.c | . . . . . 6 ⊢ 𝐶 ∈ ℕ0 | |
12 | 11 | nn0rei 11911 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
13 | dpadd2.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
14 | rpre 12400 | . . . . . 6 ⊢ (𝐷 ∈ ℝ+ → 𝐷 ∈ ℝ) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∈ ℝ |
16 | dp2cl 30560 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
17 | 12, 15, 16 | mp2an 690 | . . . 4 ⊢ _𝐶𝐷 ∈ ℝ |
18 | 10, 17 | dpval2 30573 | . . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10)) |
19 | 9, 18 | oveq12i 7171 | . 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) |
20 | 1 | nn0cni 11912 | . . 3 ⊢ 𝐺 ∈ ℂ |
21 | 8 | recni 10658 | . . . 4 ⊢ _𝐴𝐵 ∈ ℂ |
22 | 10nn 12117 | . . . . 5 ⊢ ;10 ∈ ℕ | |
23 | 22 | nncni 11651 | . . . 4 ⊢ ;10 ∈ ℂ |
24 | 22 | nnne0i 11680 | . . . 4 ⊢ ;10 ≠ 0 |
25 | 21, 23, 24 | divcli 11385 | . . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
26 | 10 | nn0cni 11912 | . . 3 ⊢ 𝐻 ∈ ℂ |
27 | 17 | recni 10658 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
28 | 27, 23, 24 | divcli 11385 | . . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ |
29 | 20, 25, 26, 28 | add4i 10867 | . 2 ⊢ ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) = ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) |
30 | dpadd2.i | . . . 4 ⊢ (𝐺 + 𝐻) = 𝐼 | |
31 | 21, 27, 23, 24 | divdiri 11400 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) |
32 | dpadd2.1 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) | |
33 | dpval 30570 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
34 | 2, 6, 33 | mp2an 690 | . . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
35 | dpval 30570 | . . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
36 | 11, 15, 35 | mp2an 690 | . . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
37 | 34, 36 | oveq12i 7171 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷) |
38 | dpadd2.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
39 | dpadd2.f | . . . . . . . . 9 ⊢ 𝐹 ∈ ℝ+ | |
40 | rpre 12400 | . . . . . . . . 9 ⊢ (𝐹 ∈ ℝ+ → 𝐹 ∈ ℝ) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐹 ∈ ℝ |
42 | dpval 30570 | . . . . . . . 8 ⊢ ((𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹) | |
43 | 38, 41, 42 | mp2an 690 | . . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹 |
44 | 32, 37, 43 | 3eqtr3i 2855 | . . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹 |
45 | 44 | oveq1i 7169 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10) |
46 | 31, 45 | eqtr3i 2849 | . . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10) |
47 | 30, 46 | oveq12i 7171 | . . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10)) |
48 | 1, 10 | nn0addcli 11937 | . . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ0 |
49 | 30, 48 | eqeltrri 2913 | . . . 4 ⊢ 𝐼 ∈ ℕ0 |
50 | 38 | nn0rei 11911 | . . . . 5 ⊢ 𝐸 ∈ ℝ |
51 | dp2cl 30560 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
52 | 50, 41, 51 | mp2an 690 | . . . 4 ⊢ _𝐸𝐹 ∈ ℝ |
53 | 49, 52 | dpval2 30573 | . . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10)) |
54 | 47, 53 | eqtr4i 2850 | . 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹) |
55 | 19, 29, 54 | 3eqtri 2851 | 1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 / cdiv 11300 ℕ0cn0 11900 ;cdc 12101 ℝ+crp 12392 _cdp2 30551 .cdp 30568 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 df-rp 12393 df-dp2 30552 df-dp 30569 |
This theorem is referenced by: hgt750lemd 31923 |
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