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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 8102 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7167 | . . 3 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 7601 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onmsuc 8154 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | 2, 5 | syl5eq 2868 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = ((𝐴 ·o ∅) +o 𝐴)) |
7 | om0 8142 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
8 | 7 | oveq1d 7171 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
9 | oa0r 8163 | . 2 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2860 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∅c0 4291 Oncon0 6191 suc csuc 6193 (class class class)co 7156 ωcom 7580 1oc1o 8095 +o coa 8099 ·o comu 8100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 |
This theorem is referenced by: oe1m 8171 omword1 8199 oeordi 8213 oeoalem 8222 oeoa 8223 oeeui 8228 oaabs2 8272 infxpenc 9444 |
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