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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 12697 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2828 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
8 | 6, 7 | num0u 12695 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
9 | 0nn0 12494 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 2, 9 | num0h 12696 | . . 3 ⊢ 0 = ((𝑇 · 0) + 0) |
11 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0cni 12491 | . . . . . 6 ⊢ 𝐸 ∈ ℂ |
14 | 13 | addlidi 11409 | . . . . 5 ⊢ (0 + 𝐸) = 𝐸 |
15 | 14 | oveq2i 7423 | . . . 4 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = ((𝐴 · 𝑃) + 𝐸) |
16 | nummul1c.8 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 | |
17 | 15, 16 | eqtri 2759 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = 𝐶 |
18 | 4, 7 | num0u 12695 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝐵 · 𝑃) + 0) |
19 | nummul1c.9 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) | |
20 | 18, 19 | eqtr3i 2761 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = ((𝑇 · 𝐸) + 𝐷) |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 12729 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ((𝑇 · 𝐶) + 𝐷) |
22 | 8, 21 | eqtri 2759 | 1 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 (class class class)co 7412 0cc0 11116 + caddc 11119 · cmul 11121 ℕ0cn0 12479 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 df-nn 12220 df-n0 12480 |
This theorem is referenced by: nummul2c 12734 decmul1c 12749 |
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