Theorem om1om1r 43297

Description: Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8580 and om1r 8581 for individual statements. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
om1om1r	⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))

Proof of Theorem om1om1r

Step	Hyp	Ref	Expression
1		om1r 8581	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2		om1 8580	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
3	1, 2	eqtr4d 2780	. 2 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = (𝐴 ·o 1o))
4	3, 2	jca 511	1 ⊢ (𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Oncon0 6384  (class class class)co 7431  1_oc1o 8499   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511

This theorem is referenced by: (None)