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Mirrors > Home > MPE Home > Th. List > Mathboxes > sq3deccom12 | Structured version Visualization version GIF version |
Description: Variant of sqdeccom12 41203 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.) |
Ref | Expression |
---|---|
sqdeccom12.a | โข ๐ด โ โ0 |
sqdeccom12.b | โข ๐ต โ โ0 |
sq3deccom12.c | โข ๐ถ โ โ0 |
sq3deccom12.d | โข (๐ด + ๐ถ) = ๐ท |
Ref | Expression |
---|---|
sq3deccom12 | โข ((;;๐ด๐ต๐ถ ยท ;;๐ด๐ต๐ถ) โ (;๐ท๐ต ยท ;๐ท๐ต)) = (;99 ยท ((;๐ด๐ต ยท ;๐ด๐ต) โ (๐ถ ยท ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq3deccom12.c | . . . . . 6 โข ๐ถ โ โ0 | |
2 | 0nn0 12486 | . . . . . 6 โข 0 โ โ0 | |
3 | sqdeccom12.a | . . . . . 6 โข ๐ด โ โ0 | |
4 | sqdeccom12.b | . . . . . 6 โข ๐ต โ โ0 | |
5 | eqid 2732 | . . . . . 6 โข ;๐ถ0 = ;๐ถ0 | |
6 | eqid 2732 | . . . . . 6 โข ;๐ด๐ต = ;๐ด๐ต | |
7 | 3 | nn0cni 12483 | . . . . . . 7 โข ๐ด โ โ |
8 | 1 | nn0cni 12483 | . . . . . . 7 โข ๐ถ โ โ |
9 | sq3deccom12.d | . . . . . . 7 โข (๐ด + ๐ถ) = ๐ท | |
10 | 7, 8, 9 | addcomli 11405 | . . . . . 6 โข (๐ถ + ๐ด) = ๐ท |
11 | 4 | nn0cni 12483 | . . . . . . 7 โข ๐ต โ โ |
12 | 11 | addlidi 11401 | . . . . . 6 โข (0 + ๐ต) = ๐ต |
13 | 1, 2, 3, 4, 5, 6, 10, 12 | decadd 12730 | . . . . 5 โข (;๐ถ0 + ;๐ด๐ต) = ;๐ท๐ต |
14 | 3, 4 | deccl 12691 | . . . . . 6 โข ;๐ด๐ต โ โ0 |
15 | 14 | nn0cni 12483 | . . . . . . 7 โข ;๐ด๐ต โ โ |
16 | 15 | addlidi 11401 | . . . . . 6 โข (0 + ;๐ด๐ต) = ;๐ด๐ต |
17 | 1, 2, 14, 5, 16 | decaddi 12736 | . . . . 5 โข (;๐ถ0 + ;๐ด๐ต) = ;๐ถ;๐ด๐ต |
18 | 13, 17 | eqtr3i 2762 | . . . 4 โข ;๐ท๐ต = ;๐ถ;๐ด๐ต |
19 | 18, 18 | oveq12i 7420 | . . 3 โข (;๐ท๐ต ยท ;๐ท๐ต) = (;๐ถ;๐ด๐ต ยท ;๐ถ;๐ด๐ต) |
20 | 19 | oveq2i 7419 | . 2 โข ((;;๐ด๐ต๐ถ ยท ;;๐ด๐ต๐ถ) โ (;๐ท๐ต ยท ;๐ท๐ต)) = ((;;๐ด๐ต๐ถ ยท ;;๐ด๐ต๐ถ) โ (;๐ถ;๐ด๐ต ยท ;๐ถ;๐ด๐ต)) |
21 | 14, 1 | sqdeccom12 41203 | . 2 โข ((;;๐ด๐ต๐ถ ยท ;;๐ด๐ต๐ถ) โ (;๐ถ;๐ด๐ต ยท ;๐ถ;๐ด๐ต)) = (;99 ยท ((;๐ด๐ต ยท ;๐ด๐ต) โ (๐ถ ยท ๐ถ))) |
22 | 20, 21 | eqtri 2760 | 1 โข ((;;๐ด๐ต๐ถ ยท ;;๐ด๐ต๐ถ) โ (;๐ท๐ต ยท ;๐ท๐ต)) = (;99 ยท ((;๐ด๐ต ยท ;๐ด๐ต) โ (๐ถ ยท ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 (class class class)co 7408 0cc0 11109 + caddc 11112 ยท cmul 11114 โ cmin 11443 9c9 12273 โ0cn0 12471 ;cdc 12676 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-ltxr 11252 df-sub 11445 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-dec 12677 |
This theorem is referenced by: (None) |
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