Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nnm1 | Structured version Visualization version GIF version |
Description: Multiply an element of ω by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnm1 | ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 7560 | . . 3 ⊢ 1𝑜 = suc ∅ | |
2 | 1 | oveq2i 6661 | . 2 ⊢ (𝐴 ·𝑜 1𝑜) = (𝐴 ·𝑜 suc ∅) |
3 | peano1 7085 | . . . 4 ⊢ ∅ ∈ ω | |
4 | nnmsuc 7687 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 suc ∅) = ((𝐴 ·𝑜 ∅) +𝑜 𝐴)) | |
5 | 3, 4 | mpan2 707 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 suc ∅) = ((𝐴 ·𝑜 ∅) +𝑜 𝐴)) |
6 | nnm0 7685 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅) | |
7 | 6 | oveq1d 6665 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 𝐴) = (∅ +𝑜 𝐴)) |
8 | nna0r 7689 | . . 3 ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴) | |
9 | 5, 7, 8 | 3eqtrd 2660 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 suc ∅) = 𝐴) |
10 | 2, 9 | syl5eq 2668 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∅c0 3915 suc csuc 5725 (class class class)co 6650 ωcom 7065 1𝑜c1o 7553 +𝑜 coa 7557 ·𝑜 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: nnm2 7729 mulidpi 9708 |
Copyright terms: Public domain | W3C validator |