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Mirrors > Home > ILE Home > Th. List > omv2 | Unicode version |
Description: Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Ref | Expression |
---|---|
omv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omfnex 6052 | . . . 4 | |
2 | 0elon 4147 | . . . . 5 | |
3 | rdgival 5992 | . . . . 5 | |
4 | 2, 3 | mp3an2 1256 | . . . 4 |
5 | 1, 4 | sylan 277 | . . 3 |
6 | omv 6058 | . . 3 | |
7 | onelon 4139 | . . . . . . 7 | |
8 | omexg 6054 | . . . . . . . . 9 | |
9 | omcl 6064 | . . . . . . . . . 10 | |
10 | simpl 107 | . . . . . . . . . 10 | |
11 | oacl 6063 | . . . . . . . . . 10 | |
12 | 9, 10, 11 | syl2anc 403 | . . . . . . . . 9 |
13 | oveq1 5539 | . . . . . . . . . 10 | |
14 | eqid 2081 | . . . . . . . . . 10 | |
15 | 13, 14 | fvmptg 5269 | . . . . . . . . 9 |
16 | 8, 12, 15 | syl2anc 403 | . . . . . . . 8 |
17 | omv 6058 | . . . . . . . . 9 | |
18 | 17 | fveq2d 5202 | . . . . . . . 8 |
19 | 16, 18 | eqtr3d 2115 | . . . . . . 7 |
20 | 7, 19 | sylan2 280 | . . . . . 6 |
21 | 20 | anassrs 392 | . . . . 5 |
22 | 21 | iuneq2dv 3699 | . . . 4 |
23 | 22 | uneq2d 3126 | . . 3 |
24 | 5, 6, 23 | 3eqtr4d 2123 | . 2 |
25 | uncom 3116 | . . 3 | |
26 | un0 3278 | . . 3 | |
27 | 25, 26 | eqtri 2101 | . 2 |
28 | 24, 27 | syl6eq 2129 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 cun 2971 c0 3251 ciun 3678 cmpt 3839 con0 4118 wfn 4917 cfv 4922 (class class class)co 5532 crdg 5979 coa 6021 comu 6022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 |
This theorem is referenced by: omsuc 6074 |
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