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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 11760 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 9nn0 11769 | . . 3 ⊢ 9 ∈ ℕ0 | |
3 | 1nn0 11761 | . . 3 ⊢ 1 ∈ ℕ0 | |
4 | 2 | dec0h 11969 | . . . . . 6 ⊢ 9 = ;09 |
5 | 4 | eqcomi 2804 | . . . . 5 ⊢ ;09 = 9 |
6 | 5 | deceq1i 11954 | . . . 4 ⊢ ;;099 = ;99 |
7 | 1 | dec0h 11969 | . . . . . 6 ⊢ 0 = ;00 |
8 | 7 | eqcomi 2804 | . . . . 5 ⊢ ;00 = 0 |
9 | 8 | deceq1i 11954 | . . . 4 ⊢ ;;001 = ;01 |
10 | 9cn 11585 | . . . . . . 7 ⊢ 9 ∈ ℂ | |
11 | 10 | addid1i 10674 | . . . . . 6 ⊢ (9 + 0) = 9 |
12 | 11 | oveq1i 7026 | . . . . 5 ⊢ ((9 + 0) + 1) = (9 + 1) |
13 | 9p1e10 11949 | . . . . 5 ⊢ (9 + 1) = ;10 | |
14 | 12, 13 | eqtri 2819 | . . . 4 ⊢ ((9 + 0) + 1) = ;10 |
15 | 2, 2, 1, 3, 6, 9, 14, 1, 13 | decaddc 12002 | . . 3 ⊢ (;;099 + ;;001) = ;;100 |
16 | 1, 2, 2, 1, 1, 3, 3, 1, 1, 15 | dpadd3 30272 | . 2 ⊢ ((0._99) + (0._01)) = (1._00) |
17 | 1 | dp20u 30238 | . . 3 ⊢ _00 = 0 |
18 | 17 | oveq2i 7027 | . 2 ⊢ (1._00) = (1.0) |
19 | 3 | dp0u 30261 | . 2 ⊢ (1.0) = 1 |
20 | 16, 18, 19 | 3eqtri 2823 | 1 ⊢ ((0._99) + (0._01)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 (class class class)co 7016 0cc0 10383 1c1 10384 + caddc 10386 9c9 11547 ;cdc 11947 _cdp2 30231 .cdp 30248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-dec 11948 df-dp2 30232 df-dp 30249 |
This theorem is referenced by: (None) |
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