Theorem nmuladdss 36656

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 6402	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ⊆ 𝐴 ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴)))
2	1	ancoms 463	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐴 ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴)))
3		onsseleq 6402	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐵 ∈ On) → (𝐷 ⊆ 𝐵 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵)))
4	3	ancoms 463	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ⊆ 𝐵 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵)))
5	2, 4	bi2anan9 649	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) ∧ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵))))
6	5	an4s 672	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 = 𝐴) ∧ (𝐷 ∈ 𝐵 ∨ 𝐷 = 𝐵))))
7		nmulcl 36649	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
8	7	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐴 ·no 𝐵) ∈ On)
9		onelon 6385	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
10		onelon 6385	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ On)
11		nmulcl 36649	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ·no 𝐷) ∈ On)
12	9, 10, 11	syl2an 607	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐷 ∈ 𝐵)) → (𝐶 ·no 𝐷) ∈ On)
13	12	an4s 672	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶 ·no 𝐷) ∈ On)
14	8, 13	naddcld 8665	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) ∈ On)
15		ontr 6472	. . . . . . . . . . 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 36655	. . . . . . . . . 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 5227	. . . . . . . . . 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 727	. . . . . . . 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 461	. . . . . . 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 786	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐴 ∈ On)
23		simpllr 787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On)
24	22, 23	nmulcld 36651	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐴 ·no 𝐵) ∈ On)
25		simplrl 788	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
26	25, 23	nmulcld 36651	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝐶 ∈ 𝐴) → (𝐶 ·no 𝐵) ∈ On)
27		naddcom 8668	. . . . . . . . . 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 595	. . . . . . . . 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 3993	. . . . . . . 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 7418	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → (𝐴 ·no 𝐷) = (𝐴 ·no 𝐵))
31	30	oveq2d 7426	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) = ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐵)))
32		oveq2 7418	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → (𝐶 ·no 𝐷) = (𝐶 ·no 𝐵))
33	32	oveq2d 7426	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐵)))
34	31, 33	sseq12d 3969	. . . . . . . 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 872	. . . . . 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 417	. . . . 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 3958	. . . . . . 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 7417	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐶 ·no 𝐵) = (𝐴 ·no 𝐵))
41	40	oveq1d 7425	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)))
42		oveq1 7417	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐶 ·no 𝐷) = (𝐴 ·no 𝐷))
43	42	oveq2d 7426	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐷)))
44	41, 43	sseq12d 3969	. . . . . . 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 872	. . . 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 415	. . 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 1133	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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ⊆ wss 3904  Tr wtr 5217  Oncon0 6360  (class class class)co 7410   +no cnadd 8650   ·_no cnmul 36645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  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 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-frecs 8277  df-nadd 8651  df-nmul 36646

This theorem is referenced by:  nmulss1  36657