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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 8505 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7442 | . . 3 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 7911 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onmsuc 8566 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | 2, 5 | eqtrid 2787 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = ((𝐴 ·o ∅) +o 𝐴)) |
7 | om0 8554 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
8 | 7 | oveq1d 7446 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
9 | oa0r 8575 | . 2 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2779 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∅c0 4339 Oncon0 6386 suc csuc 6388 (class class class)co 7431 ωcom 7887 1oc1o 8498 +o coa 8502 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 |
This theorem is referenced by: oe1m 8582 omword1 8610 oeordi 8624 oeoalem 8633 oeoa 8634 oeeui 8639 oaabs2 8686 infxpenc 10056 om1om1r 43274 oaabsb 43284 oaomoencom 43307 cantnfresb 43314 omabs2 43322 omcl3g 43324 om2 43394 |
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