Theorem nmuladdss 37140

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

Assertion

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

Proof of Theorem nmuladdss

Step	Hyp	Ref	Expression
1		onsseleq 6422	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ⊆ 𝐴 ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴)))
2	1	ancoms 464	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐴 ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴)))
3		onsseleq 6422	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐵 ∈ On) → (𝐷 ⊆ 𝐵 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵)))
4	3	ancoms 464	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ⊆ 𝐵 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵)))
5	2, 4	bi2anan9 650	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) ∧ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵))))
6	5	an4s 673	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) ∧ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵))))
7		nmulcl 37118	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
8	7	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐴 ·no 𝐵) ∈ On)
9		onelon 6405	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
10		onelon 6405	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ On)
11		nmulcl 37118	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ·no 𝐷) ∈ On)
12	9, 10, 11	syl2an 608	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐷 ∈ 𝐵)) → (𝐶 ·no 𝐷) ∈ On)
13	12	an4s 673	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶 ·no 𝐷) ∈ On)
14	8, 13	naddcld 8717	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) ∈ On)
15		ontr 6492	. . . . . . . . . . 11 ⊢ (((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) ∈ On → Tr ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
16	14, 15	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → Tr ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
17		nmuladdel 37139	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
18		trss 5225	. . . . . . . . . 10 ⊢ (Tr ((𝐴 ·no 𝐵) +no (𝐶 ·no 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))))
19	16, 17, 18	sylc 66	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
20	19	adantlr 728	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
21	20	expr 462	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
22		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐴 ∈ On)
23		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On)
24	22, 23	nmulcld 37120	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐴 ·no 𝐵) ∈ On)
25		simplrl 789	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
26	25, 23	nmulcld 37120	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐶 ·no 𝐵) ∈ On)
27		naddcom 8720	. . . . . . . . . 10 ⊢ (((𝐴 ·no 𝐵) ∈ On ∧ (𝐶 ·no 𝐵) ∈ On) → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)))
28	24, 26, 27	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)))
29	28	eqimsscd 3970	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵)))
30		oveq2 7451	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → (𝐴 ·no 𝐷) = (𝐴 ·no 𝐵))
31	30	oveq2d 7459	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)))
32		oveq2 7451	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → (𝐶 ·no 𝐷) = (𝐶 ·no 𝐵))
33	32	oveq2d 7459	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵)))
34	31, 33	sseq12d 3946	. . . . . . . 8 ⊢ (𝐷 = 𝐵 → (((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) ↔ ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵))))
35	29, 34	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐷 = 𝐵 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
36	21, 35	jaod 873	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
37	36	ex 418	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ 𝐴 → ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))))
38		ssid 3935	. . . . . . 7 ⊢ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷))
39	38	2a1i 12	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷))))
40		oveq1 7450	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐶 ·no 𝐵) = (𝐴 ·no 𝐵))
41	40	oveq1d 7458	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)))
42		oveq1 7450	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐶 ·no 𝐷) = (𝐴 ·no 𝐷))
43	42	oveq2d 7459	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)))
44	41, 43	sseq12d 3946	. . . . . . 7 ⊢ (𝐶 = 𝐴 → (((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) ↔ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷))))
45	44	imbi2d 343	. . . . . 6 ⊢ (𝐶 = 𝐴 → (((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))) ↔ ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)))))
46	39, 45	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 = 𝐴 → ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))))
47	37, 46	jaod 873	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) → ((𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))))
48	47	impd 416	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) ∧ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
49	6, 48	sylbid 243	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷))))
50	49	3impia 1135	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ 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   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  Tr wtr 5215  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:  nmulss1  37141  nadddilem3  37149