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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1mhdrd | Structured version Visualization version GIF version |
Description: Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
Ref | Expression |
---|---|
1mhdrd | ⊢ ((0._99) + (0._01)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12483 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 9nn0 12492 | . . 3 ⊢ 9 ∈ ℕ0 | |
3 | 1nn0 12484 | . . 3 ⊢ 1 ∈ ℕ0 | |
4 | 2 | dec0h 12695 | . . . . . 6 ⊢ 9 = ;09 |
5 | 4 | eqcomi 2733 | . . . . 5 ⊢ ;09 = 9 |
6 | 5 | deceq1i 12680 | . . . 4 ⊢ ;;099 = ;99 |
7 | 1 | dec0h 12695 | . . . . . 6 ⊢ 0 = ;00 |
8 | 7 | eqcomi 2733 | . . . . 5 ⊢ ;00 = 0 |
9 | 8 | deceq1i 12680 | . . . 4 ⊢ ;;001 = ;01 |
10 | 9cn 12308 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
11 | 10 | addridi 11397 | . . . . . 6 ⊢ (9 + 0) = 9 |
12 | 11 | oveq1i 7411 | . . . . 5 ⊢ ((9 + 0) + 1) = (9 + 1) |
13 | 9p1e10 12675 | . . . . 5 ⊢ (9 + 1) = ;10 | |
14 | 12, 13 | eqtri 2752 | . . . 4 ⊢ ((9 + 0) + 1) = ;10 |
15 | 2, 2, 1, 3, 6, 9, 14, 1, 13 | decaddc 12728 | . . 3 ⊢ (;;099 + ;;001) = ;;100 |
16 | 1, 2, 2, 1, 1, 3, 3, 1, 1, 15 | dpadd3 32511 | . 2 ⊢ ((0._99) + (0._01)) = (1._00) |
17 | 1 | dp20u 32477 | . . 3 ⊢ _00 = 0 |
18 | 17 | oveq2i 7412 | . 2 ⊢ (1._00) = (1.0) |
19 | 3 | dp0u 32500 | . 2 ⊢ (1.0) = 1 |
20 | 16, 18, 19 | 3eqtri 2756 | 1 ⊢ ((0._99) + (0._01)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7401 0cc0 11105 1c1 11106 + caddc 11108 9c9 12270 ;cdc 12673 _cdp2 32470 .cdp 32487 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-dec 12674 df-dp2 32471 df-dp 32488 |
This theorem is referenced by: (None) |
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