omopth2 - Metamath Proof Explorer

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Theorem omopth2 7664

Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)

**Assertion**
Ref	Expression
omopth2	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem omopth2**
Step	Hyp	Ref	Expression
1		simpl2l 1114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		eloni 5733	. . . . . . 7
3	1, 2	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simpl3l 1116	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		eloni 5733	. . . . . . 7
6	4, 5	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		ordtri3or 5755	. . . . . 6 $/\$ $\/$ $\/$
8	3, 6, 7	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
9		simpr 477	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpl1l 1112	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11		omcl 7616	. . . . . . . . . . . 12 $/\$
12	10, 4, 11	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simpl3r 1117	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		onelon 5748	. . . . . . . . . . . 12 $/\$
15	10, 13, 14	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		oacl 7615	. . . . . . . . . . 11 $/\$
17	12, 15, 16	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		eloni 5733	. . . . . . . . . 10
19		ordirr 5741	. . . . . . . . . 10
20	17, 18, 19	3syl 18	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 20	eqneltrd 2720	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		orc 400	. . . . . . . . 9 $\/$ $/\$
23		omeulem2 7663	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
24	23	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
25	22, 24	syl5 34	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	21, 25	mtod 189	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	26	pm2.21d 118	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		idd 24	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	20, 9	neleqtrrd 2723	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		orc 400	. . . . . . . . 9 $\/$ $/\$
31		simpl1r 1113	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simpl2r 1115	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		omeulem2 7663	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
34	10, 31, 4, 13, 1, 32, 33	syl222anc 1342	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
35	30, 34	syl5 34	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	29, 35	mtod 189	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	36	pm2.21d 118	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	27, 28, 37	3jaod 1392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
39	8, 38	mpd 15	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		onelon 5748	. . . . . . . 8 $/\$
41		eloni 5733	. . . . . . . 8
42	40, 41	syl 17	. . . . . . 7 $/\$
43	10, 32, 42	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		eloni 5733	. . . . . . . 8
45	14, 44	syl 17	. . . . . . 7 $/\$
46	10, 13, 45	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		ordtri3or 5755	. . . . . 6 $/\$ $\/$ $\/$
48	43, 46, 47	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
49		olc 399	. . . . . . . . . 10 $/\$ $\/$ $/\$
50	49, 24	syl5 34	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	39, 50	mpand 711	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	21, 51	mtod 189	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52	pm2.21d 118	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		idd 24	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	39	eqcomd 2628	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		olc 399	. . . . . . . . . 10 $/\$ $\/$ $/\$
57	56, 34	syl5 34	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	55, 57	mpand 711	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	29, 58	mtod 189	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	pm2.21d 118	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	53, 54, 60	3jaod 1392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
62	48, 61	mpd 15	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	39, 62	jca 554	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	63	ex 450	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		oveq2 6658	. . 3
66		id 22	. . 3
67	65, 66	oveqan12d 6669	. 2 $/\$
68	64, 67	impbid1 215	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

c0 3915

word 5722

con0 5723 (class class class)co 6650

coa 7557

comu 7558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-omul 7565

This theorem is referenced by: omeu 7665 dfac12lem2 8966

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