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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. Lemma 2.15 of [Schloeder] p. 5. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
om1 | ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8491 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7435 | . . 3 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 7898 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onmsuc 8554 | . . . 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 | eqtrid 2779 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = ((𝐴 ·o ∅) +o 𝐴)) |
7 | om0 8542 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
8 | 7 | oveq1d 7439 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
9 | oa0r 8563 | . 2 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2771 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4324 Oncon0 6372 suc csuc 6374 (class class class)co 7424 ωcom 7874 1oc1o 8484 +o coa 8488 ·o comu 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-oadd 8495 df-omul 8496 |
This theorem is referenced by: oe1m 8570 omword1 8598 oeordi 8612 oeoalem 8621 oeoa 8622 oeeui 8627 oaabs2 8674 infxpenc 10047 om1om1r 42716 oaabsb 42726 oaomoencom 42749 cantnfresb 42756 omabs2 42764 omcl3g 42766 om2 42837 |
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