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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 8409 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7365 | . 2 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 7822 | . . . 4 ⊢ ∅ ∈ ω | |
4 | nnmsuc 8551 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | nnm0 8549 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | |
7 | 6 | oveq1d 7369 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
8 | nna0r 8553 | . . 3 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | |
9 | 5, 7, 8 | 3eqtrd 2780 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = 𝐴) |
10 | 2, 9 | eqtrid 2788 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∅c0 4281 suc csuc 6318 (class class class)co 7354 ωcom 7799 1oc1o 8402 +o coa 8406 ·o comu 8407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-oadd 8413 df-omul 8414 |
This theorem is referenced by: nnm2 8596 mulidpi 10819 |
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