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Mirrors > Home > ILE Home > Th. List > mulpiord | Unicode version |
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
mulpiord |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4376 | . 2 | |
2 | fvres 5198 | . . 3 | |
3 | df-ov 5515 | . . . 4 | |
4 | df-mi 6404 | . . . . 5 | |
5 | 4 | fveq1i 5179 | . . . 4 |
6 | 3, 5 | eqtri 2060 | . . 3 |
7 | df-ov 5515 | . . 3 | |
8 | 2, 6, 7 | 3eqtr4g 2097 | . 2 |
9 | 1, 8 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cop 3378 cxp 4343 cres 4347 cfv 4902 (class class class)co 5512 comu 5999 cnpi 6370 cmi 6372 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-res 4357 df-iota 4867 df-fv 4910 df-ov 5515 df-mi 6404 |
This theorem is referenced by: mulidpi 6416 mulclpi 6426 mulcompig 6429 mulasspig 6430 distrpig 6431 mulcanpig 6433 ltmpig 6437 archnqq 6515 enq0enq 6529 addcmpblnq0 6541 mulcmpblnq0 6542 mulcanenq0ec 6543 addclnq0 6549 mulclnq0 6550 nqpnq0nq 6551 nqnq0a 6552 nqnq0m 6553 nq0m0r 6554 distrnq0 6557 addassnq0lemcl 6559 |
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