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Mirrors > Home > MPE Home > Th. List > 9t11e99 | Structured version Visualization version GIF version |
Description: 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
9t11e99 | ⊢ (9 · ;11) = ;99 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9cn 11296 | . . . 4 ⊢ 9 ∈ ℂ | |
2 | 10nn0 11704 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
3 | 2 | nn0cni 11492 | . . . . 5 ⊢ ;10 ∈ ℂ |
4 | ax-1cn 10182 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | mulcli 10233 | . . . 4 ⊢ (;10 · 1) ∈ ℂ |
6 | 1, 5, 4 | adddii 10238 | . . 3 ⊢ (9 · ((;10 · 1) + 1)) = ((9 · (;10 · 1)) + (9 · 1)) |
7 | 3 | mulid1i 10230 | . . . . . 6 ⊢ (;10 · 1) = ;10 |
8 | 7 | oveq2i 6820 | . . . . 5 ⊢ (9 · (;10 · 1)) = (9 · ;10) |
9 | 1, 3 | mulcomi 10234 | . . . . 5 ⊢ (9 · ;10) = (;10 · 9) |
10 | 8, 9 | eqtri 2778 | . . . 4 ⊢ (9 · (;10 · 1)) = (;10 · 9) |
11 | 1 | mulid1i 10230 | . . . 4 ⊢ (9 · 1) = 9 |
12 | 10, 11 | oveq12i 6821 | . . 3 ⊢ ((9 · (;10 · 1)) + (9 · 1)) = ((;10 · 9) + 9) |
13 | 6, 12 | eqtri 2778 | . 2 ⊢ (9 · ((;10 · 1) + 1)) = ((;10 · 9) + 9) |
14 | dfdec10 11685 | . . 3 ⊢ ;11 = ((;10 · 1) + 1) | |
15 | 14 | oveq2i 6820 | . 2 ⊢ (9 · ;11) = (9 · ((;10 · 1) + 1)) |
16 | dfdec10 11685 | . 2 ⊢ ;99 = ((;10 · 9) + 9) | |
17 | 13, 15, 16 | 3eqtr4i 2788 | 1 ⊢ (9 · ;11) = ;99 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1628 (class class class)co 6809 0cc0 10124 1c1 10125 + caddc 10127 · cmul 10129 9c9 11265 ;cdc 11681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-ov 6812 df-om 7227 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-er 7907 df-en 8118 df-dom 8119 df-sdom 8120 df-pnf 10264 df-mnf 10265 df-ltxr 10267 df-nn 11209 df-2 11267 df-3 11268 df-4 11269 df-5 11270 df-6 11271 df-7 11272 df-8 11273 df-9 11274 df-n0 11481 df-dec 11682 |
This theorem is referenced by: 3dvds2dec 15254 3dvds2decOLD 15255 1259lem3 16038 |
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