Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6338 | . . . 4 | |
2 | 0elon 4309 | . . . . 5 | |
3 | rdgival 6272 | . . . . 5 | |
4 | 2, 3 | mp3an2 1303 | . . . 4 |
5 | 1, 4 | sylan 281 | . . 3 |
6 | omv 6344 | . . 3 | |
7 | onelon 4301 | . . . . . . 7 | |
8 | omexg 6340 | . . . . . . . . 9 | |
9 | omcl 6350 | . . . . . . . . . 10 | |
10 | simpl 108 | . . . . . . . . . 10 | |
11 | oacl 6349 | . . . . . . . . . 10 | |
12 | 9, 10, 11 | syl2anc 408 | . . . . . . . . 9 |
13 | oveq1 5774 | . . . . . . . . . 10 | |
14 | eqid 2137 | . . . . . . . . . 10 | |
15 | 13, 14 | fvmptg 5490 | . . . . . . . . 9 |
16 | 8, 12, 15 | syl2anc 408 | . . . . . . . 8 |
17 | omv 6344 | . . . . . . . . 9 | |
18 | 17 | fveq2d 5418 | . . . . . . . 8 |
19 | 16, 18 | eqtr3d 2172 | . . . . . . 7 |
20 | 7, 19 | sylan2 284 | . . . . . 6 |
21 | 20 | anassrs 397 | . . . . 5 |
22 | 21 | iuneq2dv 3829 | . . . 4 |
23 | 22 | uneq2d 3225 | . . 3 |
24 | 5, 6, 23 | 3eqtr4d 2180 | . 2 |
25 | uncom 3215 | . . 3 | |
26 | un0 3391 | . . 3 | |
27 | 25, 26 | eqtri 2158 | . 2 |
28 | 24, 27 | syl6eq 2186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2681 cun 3064 c0 3358 ciun 3808 cmpt 3984 con0 4280 wfn 5113 cfv 5118 (class class class)co 5767 crdg 6259 coa 6303 comu 6304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 |
This theorem is referenced by: omsuc 6361 |
Copyright terms: Public domain | W3C validator |