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Mirrors > Home > ILE Home > Th. List > 11multnc | GIF version |
Description: The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.) |
Ref | Expression |
---|---|
11multnc.n | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
11multnc | ⊢ (𝑁 · ;11) = ;𝑁𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 11multnc.n | . . 3 ⊢ 𝑁 ∈ ℕ0 | |
2 | 1nn0 9205 | . . 3 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2, 2 | decmulnc 9463 | . 2 ⊢ (𝑁 · ;11) = ;(𝑁 · 1)(𝑁 · 1) |
4 | 1 | nn0cni 9201 | . . . 4 ⊢ 𝑁 ∈ ℂ |
5 | 4 | mulid1i 7972 | . . 3 ⊢ (𝑁 · 1) = 𝑁 |
6 | 5, 5 | deceq12i 9405 | . 2 ⊢ ;(𝑁 · 1)(𝑁 · 1) = ;𝑁𝑁 |
7 | 3, 6 | eqtri 2208 | 1 ⊢ (𝑁 · ;11) = ;𝑁𝑁 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2158 (class class class)co 5888 1c1 7825 · cmul 7829 ℕ0cn0 9189 ;cdc 9397 |
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 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 |
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 8143 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-dec 9398 |
This theorem is referenced by: (None) |
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