Theorem nmulfn 36756

Description: Natural multiplication is a function over pairs of ordinals. (Contributed by Scott Fenton, 2-Jun-2026.)

Assertion

Ref	Expression
nmulfn	⊢ ·no Fn (On × On)

Proof of Theorem nmulfn

Dummy variables 𝑥 𝑦 𝑧 𝑝 𝑚 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nmul 36755	. 2 ⊢ ·no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑝 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑎⦌⦋(2nd ‘𝑝) / 𝑏⦌∩ {𝑧 ∈ On ∣ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 ((𝑐𝑚𝑏) +no (𝑎𝑚𝑑)) ∈ (𝑧 +no (𝑐𝑚𝑑))}))
2	1	on2recsfn 8658	1 ⊢ ·no Fn (On × On)

This proof depends on syntax axioms:   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453  ⦋csb 3850  ∩ cint 4910   × cxp 5657  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7987  2^nd c2nd 7988   +no cnadd 8656   ·_no cnmul 36754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-1st 7989  df-2nd 7990  df-frecs 8283  df-nmul 36755

This theorem is used by:  nmulprop  36757