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Mirrors > Home > MPE Home > Th. List > om1 | Structured version Visualization version GIF version |
Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
om1 | ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8085 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7146 | . . 3 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 7581 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onmsuc 8137 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | 2, 5 | syl5eq 2845 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = ((𝐴 ·o ∅) +o 𝐴)) |
7 | om0 8125 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
8 | 7 | oveq1d 7150 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
9 | oa0r 8146 | . 2 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2837 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∅c0 4243 Oncon0 6159 suc csuc 6161 (class class class)co 7135 ωcom 7560 1oc1o 8078 +o coa 8082 ·o comu 8083 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 |
This theorem is referenced by: oe1m 8154 omword1 8182 oeordi 8196 oeoalem 8205 oeoa 8206 oeeui 8211 oaabs2 8255 infxpenc 9429 |
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