Theorem nmuladdel 36779

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 7423	. . . . 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 36759	. . . 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 4031	. . . . 5 ⊢ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ⊆ On
6		nmulcl 36758	. . . . . . 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 4342	. . . . . . 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 7799	. . . . . . 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 7792	. . . . 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 599	. . . 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 3651	. 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 7423	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝐵) = (𝐶 ·no 𝐵))
17	16	oveq1d 7431	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)))
18		oveq1 7423	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝑑) = (𝐶 ·no 𝑑))
19	18	oveq2d 7432	. . . . 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 7424	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐴 ·no 𝑑) = (𝐴 ·no 𝐷))
22	21	oveq2d 7432	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)))
23		oveq2 7424	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 ·no 𝑑) = (𝐶 ·no 𝐷))
24	23	oveq2d 7432	. . . . 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 3591	. . 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 464	. 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 592	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 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ⊆ wss 3902  ∅c0 4282  ∩ cint 4910  Oncon0 6361  (class class class)co 7416   +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-int 4911  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-suc 6367  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-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-frecs 8283  df-nadd 8657  df-nmul 36755

This theorem is used by:  nmuladdss  36780  nmulel1  36782  ltnmul  36783  nadddilem1  36787  nadddilem3  36789