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Mirrors > Home > MPE Home > Th. List > nummul1c | Structured version Visualization version GIF version |
Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 | ⊢ 𝑇 ∈ ℕ0 |
nummul1c.2 | ⊢ 𝑃 ∈ ℕ0 |
nummul1c.3 | ⊢ 𝐴 ∈ ℕ0 |
nummul1c.4 | ⊢ 𝐵 ∈ ℕ0 |
nummul1c.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) |
nummul1c.6 | ⊢ 𝐷 ∈ ℕ0 |
nummul1c.7 | ⊢ 𝐸 ∈ ℕ0 |
nummul1c.8 | ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
nummul1c.9 | ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
Ref | Expression |
---|---|
nummul1c | ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) | |
2 | nummul1c.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
3 | nummul1c.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
4 | nummul1c.4 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
5 | 2, 3, 4 | numcl 12112 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2909 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
8 | 6, 7 | num0u 12110 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
9 | 0nn0 11913 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 2, 9 | num0h 12111 | . . 3 ⊢ 0 = ((𝑇 · 0) + 0) |
11 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0cni 11910 | . . . . . 6 ⊢ 𝐸 ∈ ℂ |
14 | 13 | addid2i 10828 | . . . . 5 ⊢ (0 + 𝐸) = 𝐸 |
15 | 14 | oveq2i 7167 | . . . 4 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = ((𝐴 · 𝑃) + 𝐸) |
16 | nummul1c.8 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 | |
17 | 15, 16 | eqtri 2844 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = 𝐶 |
18 | 4, 7 | num0u 12110 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝐵 · 𝑃) + 0) |
19 | nummul1c.9 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) | |
20 | 18, 19 | eqtr3i 2846 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = ((𝑇 · 𝐸) + 𝐷) |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 12144 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ((𝑇 · 𝐶) + 𝐷) |
22 | 8, 21 | eqtri 2844 | 1 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 0cc0 10537 + caddc 10540 · cmul 10542 ℕ0cn0 11898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-n0 11899 |
This theorem is referenced by: nummul2c 12149 decmul1c 12164 |
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