Theorem nmuladdel 37003

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 7427	. . . . 5 ⊢ (𝑥 = (𝐴 ·no 𝐵) → (𝑥 +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)))
2	1	eleq2d 2846	. . . 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 3226	. . 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 36983	. . . 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 4005	. . . . 5 ⊢ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ⊆ On
6		nmulcl 36982	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
7	4, 6	eqeltrrd 2861	. . . . . 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 4338	. . . . . . 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 7811	. . . . . . 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 7804	. . . . 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 2860	. . 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 3647	. 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 7427	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝐵) = (𝐶 ·no 𝐵))
17	16	oveq1d 7435	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)))
18		oveq1 7427	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝑑) = (𝐶 ·no 𝑑))
19	18	oveq2d 7436	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)))
20	17, 19	eleq12d 2854	. . . 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 7428	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐴 ·no 𝑑) = (𝐴 ·no 𝐷))
22	21	oveq2d 7436	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)))
23		oveq2 7428	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 ·no 𝑑) = (𝐶 ·no 𝐷))
24	23	oveq2d 7436	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
25	22, 24	eleq12d 2854	. . . 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 3587	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895  ∅c0 4278  ∩ cint 4906  Oncon0 6362  (class class class)co 7420   +no cnadd 8674   ·_no cnmul 36978

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-nadd 8675  df-nmul 36979

This theorem is used by:  nmuladdss  37004  nmulel1  37006  ltnmul  37007  nadddilem1  37011  nadddilem3  37013