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Mirrors > Home > ILE Home > Th. List > 9t11e99 | 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 9021 | . . . 4 ⊢ 9 ∈ ℂ | |
2 | 10nn0 9415 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
3 | 2 | nn0cni 9202 | . . . . 5 ⊢ ;10 ∈ ℂ |
4 | ax-1cn 7918 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | mulcli 7976 | . . . 4 ⊢ (;10 · 1) ∈ ℂ |
6 | 1, 5, 4 | adddii 7981 | . . 3 ⊢ (9 · ((;10 · 1) + 1)) = ((9 · (;10 · 1)) + (9 · 1)) |
7 | 3 | mulid1i 7973 | . . . . . 6 ⊢ (;10 · 1) = ;10 |
8 | 7 | oveq2i 5899 | . . . . 5 ⊢ (9 · (;10 · 1)) = (9 · ;10) |
9 | 1, 3 | mulcomi 7977 | . . . . 5 ⊢ (9 · ;10) = (;10 · 9) |
10 | 8, 9 | eqtri 2208 | . . . 4 ⊢ (9 · (;10 · 1)) = (;10 · 9) |
11 | 1 | mulid1i 7973 | . . . 4 ⊢ (9 · 1) = 9 |
12 | 10, 11 | oveq12i 5900 | . . 3 ⊢ ((9 · (;10 · 1)) + (9 · 1)) = ((;10 · 9) + 9) |
13 | 6, 12 | eqtri 2208 | . 2 ⊢ (9 · ((;10 · 1) + 1)) = ((;10 · 9) + 9) |
14 | dfdec10 9401 | . . 3 ⊢ ;11 = ((;10 · 1) + 1) | |
15 | 14 | oveq2i 5899 | . 2 ⊢ (9 · ;11) = (9 · ((;10 · 1) + 1)) |
16 | dfdec10 9401 | . 2 ⊢ ;99 = ((;10 · 9) + 9) | |
17 | 13, 15, 16 | 3eqtr4i 2218 | 1 ⊢ (9 · ;11) = ;99 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 (class class class)co 5888 0cc0 7825 1c1 7826 + caddc 7828 · cmul 7830 9c9 8991 ;cdc 9398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-7 8997 df-8 8998 df-9 8999 df-n0 9191 df-dec 9399 |
This theorem is referenced by: 3dvds2dec 11885 |
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