Theorem nmuladdel 36712

Description: Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)

Assertion

Ref	Expression
nmuladdel	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))

Proof of Theorem nmuladdel

Dummy variables 𝑥 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7419	. . . . 5 ⊢ (𝑥 = (𝐴 ·no 𝐵) → (𝑥 +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)))
2	1	eleq2d 2848	. . . 4 ⊢ (𝑥 = (𝐴 ·no 𝐵) → (((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑)) ↔ ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑))))
3	2	2ralbidv 3228	. . 3 ⊢ (𝑥 = (𝐴 ·no 𝐵) → (∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑)) ↔ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑))))
4		nmulval 36692	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))})
5		ssrab2 4033	. . . . 5 ⊢ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ⊆ On
6		nmulcl 36691	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
7	4, 6	eqeltrrd 2863	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ∈ On)
8		rabn0 4345	. . . . . . 7 ⊢ ({𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ≠ ∅ ↔ ∃𝑥 ∈ On ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑)))
9		onintrab2 7794	. . . . . . 7 ⊢ (∃𝑥 ∈ On ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑)) ↔ ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ∈ On)
10	8, 9	bitri 278	. . . . . 6 ⊢ ({𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ∈ On)
11	7, 10	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ≠ ∅)
12		onint 7787	. . . . 5 ⊢ (({𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ⊆ On ∧ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ≠ ∅) → ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ∈ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))})
13	5, 11, 12	sylancr 598	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∩ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ∈ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))})
14	4, 13	eqeltrd 2862	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))})
15	3, 14	elrabrd 3652	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)))
16		oveq1 7419	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝐵) = (𝐶 ·no 𝐵))
17	16	oveq1d 7427	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)))
18		oveq1 7419	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝑑) = (𝐶 ·no 𝑑))
19	18	oveq2d 7428	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)))
20	17, 19	eleq12d 2856	. . . 4 ⊢ (𝑐 = 𝐶 → (((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)) ↔ ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑))))
21		oveq2 7420	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐴 ·no 𝑑) = (𝐴 ·no 𝐷))
22	21	oveq2d 7428	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)))
23		oveq2 7420	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 ·no 𝑑) = (𝐶 ·no 𝐷))
24	23	oveq2d 7428	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
25	22, 24	eleq12d 2856	. . . 4 ⊢ (𝑑 = 𝐷 → (((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)) ↔ ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
26	20, 25	rspc2va 3592	. . 3 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑))) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
27	26	ancoms 463	. 2 ⊢ ((∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
28	15, 27	sylan 591	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415   ⊆ wss 3904  ∅c0 4285  ∩ cint 4911  Oncon0 6360  (class class class)co 7412   +no cnadd 8649   ·_no cnmul 36687

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-frecs 8276  df-nadd 8650  df-nmul 36688

This theorem is used by:  nmuladdss  36713  nmulel1  36715  ltnmul  36716  nadddilem1  36720  nadddilem3  36722