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Mirrors > Home > MPE Home > Th. List > mulclpi | Structured version Visualization version GIF version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulclpi | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpiord 9586 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) | |
2 | pinn 9579 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
3 | pinn 9579 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
4 | nnmcl 7579 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω) | |
5 | 2, 3, 4 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 𝐵) ∈ ω) |
6 | elni2 9578 | . . . . . . 7 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
7 | 6 | simprbi 479 | . . . . . 6 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) |
9 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ ω) |
10 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ ω) |
11 | elni2 9578 | . . . . . . . 8 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | |
12 | 11 | simprbi 479 | . . . . . . 7 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐴) |
14 | nnmordi 7598 | . . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵))) | |
15 | 9, 10, 13, 14 | syl21anc 1317 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵))) |
16 | 8, 15 | mpd 15 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)) |
17 | ne0i 3880 | . . . 4 ⊢ ((𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵) → (𝐴 ·𝑜 𝐵) ≠ ∅) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 𝐵) ≠ ∅) |
19 | elni 9577 | . . 3 ⊢ ((𝐴 ·𝑜 𝐵) ∈ N ↔ ((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝐵) ≠ ∅)) | |
20 | 5, 18, 19 | sylanbrc 695 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 𝐵) ∈ N) |
21 | 1, 20 | eqeltrd 2688 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 (class class class)co 6549 ωcom 6957 ·𝑜 comu 7445 Ncnpi 9545 ·N cmi 9547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-omul 7452 df-ni 9573 df-mi 9575 |
This theorem is referenced by: mulasspi 9598 distrpi 9599 mulcanpi 9601 ltmpi 9605 enqer 9622 addpqf 9645 mulpqf 9647 adderpqlem 9655 mulerpqlem 9656 addassnq 9659 mulassnq 9660 mulcanenq 9661 distrnq 9662 recmulnq 9665 ltsonq 9670 lterpq 9671 ltanq 9672 ltmnq 9673 ltexnq 9676 archnq 9681 |
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