Theorem nmuladdel 37139

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 7450	. . . . 5 ⊢ (𝑥 = (𝐴 ·no 𝐵) → (𝑥 +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)))
2	1	eleq2d 2850	. . . 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 3230	. . 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 37119	. . . 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 4008	. . . . 5 ⊢ {𝑥 ∈ On ∣ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐵 ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) ∈ (𝑥 +no (𝑐 ·no 𝑑))} ⊆ On
6		nmulcl 37118	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
7	4, 6	eqeltrrd 2865	. . . . . 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 7835	. . . . . . 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 7828	. . . . 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 2864	. . 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 7450	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝐵) = (𝐶 ·no 𝐵))
17	16	oveq1d 7458	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)))
18		oveq1 7450	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐 ·no 𝑑) = (𝐶 ·no 𝑑))
19	18	oveq2d 7459	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no 𝐵) +no (𝑐 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)))
20	17, 19	eleq12d 2858	. . . 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 7451	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐴 ·no 𝑑) = (𝐴 ·no 𝐷))
22	21	oveq2d 7459	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝑑)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)))
23		oveq2 7451	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝐶 ·no 𝑑) = (𝐶 ·no 𝐷))
24	23	oveq2d 7459	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝑑)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
25	22, 24	eleq12d 2858	. . . 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∩ cint 4910  Oncon0 6380  (class class class)co 7443   +no cnadd 8702   ·_no cnmul 37114

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-nadd 8703  df-nmul 37115

This theorem is used by:  nmuladdss  37140  nmulel1  37142  ltnmul  37143  nadddilem1  37147  nadddilem3  37149