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Mirrors > Home > MPE Home > Th. List > nnm2 | Structured version Visualization version GIF version |
Description: Multiply an element of ω by 2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnm2 | ⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 8181 | . . 3 ⊢ 2o = suc 1o | |
2 | 1 | oveq2i 7202 | . 2 ⊢ (𝐴 ·o 2o) = (𝐴 ·o suc 1o) |
3 | 1onn 8345 | . . . 4 ⊢ 1o ∈ ω | |
4 | nnmsuc 8313 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ ω) → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) | |
5 | 3, 4 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) |
6 | nnm1 8355 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | |
7 | 6 | oveq1d 7206 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o 1o) +o 𝐴) = (𝐴 +o 𝐴)) |
8 | 5, 7 | eqtrd 2771 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = (𝐴 +o 𝐴)) |
9 | 2, 8 | syl5eq 2783 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 suc csuc 6193 (class class class)co 7191 ωcom 7622 1oc1o 8173 2oc2o 8174 +o coa 8177 ·o comu 8178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 |
This theorem is referenced by: nn2m 8357 omopthlem1 8362 omopthlem2 8363 |
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