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Mirrors > Home > ILE Home > Th. List > decmul1 | GIF version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.0 | ⊢ 𝐷 ∈ ℕ0 |
decmul1.c | ⊢ (𝐴 · 𝑃) = 𝐶 |
decmul1.d | ⊢ (𝐵 · 𝑃) = 𝐷 |
Ref | Expression |
---|---|
decmul1 | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 9432 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
3 | decmul1.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | decmul1.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 9418 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2210 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | 0nn0 9222 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 3, 2 | nn0mulcli 9245 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 9219 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 8130 | . . . 4 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
13 | decmul1.c | . . . 4 ⊢ (𝐴 · 𝑃) = 𝐶 | |
14 | 12, 13 | eqtri 2210 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = 𝐶 |
15 | decmul1.d | . . . . 5 ⊢ (𝐵 · 𝑃) = 𝐷 | |
16 | 15 | oveq2i 5908 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (0 + 𝐷) |
17 | 4, 2 | nn0mulcli 9245 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
18 | 17 | nn0cni 9219 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
19 | 18 | addid2i 8131 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (𝐵 · 𝑃) |
20 | 1 | nn0cni 9219 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
21 | 20 | mul01i 8379 | . . . . . 6 ⊢ (;10 · 0) = 0 |
22 | 21 | eqcomi 2193 | . . . . 5 ⊢ 0 = (;10 · 0) |
23 | 22 | oveq1i 5907 | . . . 4 ⊢ (0 + 𝐷) = ((;10 · 0) + 𝐷) |
24 | 16, 19, 23 | 3eqtr3i 2218 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 0) + 𝐷) |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 9463 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
26 | dfdec10 9418 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
27 | 25, 26 | eqtr4i 2213 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5897 0cc0 7842 1c1 7843 + caddc 7845 · cmul 7847 ℕ0cn0 9207 ;cdc 9415 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-dec 9416 |
This theorem is referenced by: sq10 10727 |
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