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Mirrors > Home > MPE Home > Th. List > fovcl | Structured version Visualization version Unicode version |
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) |
Ref | Expression |
---|---|
fovcl.1 |
Ref | Expression |
---|---|
fovcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovcl.1 | . . 3 | |
2 | ffnov 6764 | . . . 4 | |
3 | 2 | simprbi 480 | . . 3 |
4 | 1, 3 | ax-mp 5 | . 2 |
5 | oveq1 6657 | . . . 4 | |
6 | 5 | eleq1d 2686 | . . 3 |
7 | oveq2 6658 | . . . 4 | |
8 | 7 | eleq1d 2686 | . . 3 |
9 | 6, 8 | rspc2v 3322 | . 2 |
10 | 4, 9 | mpi 20 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cxp 5112 wfn 5883 wf 5884 (class class class)co 6650 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 |
This theorem is referenced by: addclnq 9767 mulclnq 9769 adderpq 9778 mulerpq 9779 distrnq 9783 axaddcl 9972 axmulcl 9974 xaddcl 12070 xmulcl 12103 elfzoelz 12470 addcnlem 22667 sgmcl 24872 hvaddcl 27869 hvmulcl 27870 hicl 27937 hhssabloilem 28118 rmxynorm 37483 rmxyneg 37485 rmxy1 37487 rmxy0 37488 rmxp1 37497 rmyp1 37498 rmxm1 37499 rmym1 37500 rmxluc 37501 rmyluc 37502 rmyluc2 37503 rmxdbl 37504 rmydbl 37505 rmxypos 37514 ltrmynn0 37515 ltrmxnn0 37516 lermxnn0 37517 rmxnn 37518 ltrmy 37519 rmyeq0 37520 rmyeq 37521 lermy 37522 rmynn 37523 rmynn0 37524 rmyabs 37525 jm2.24nn 37526 jm2.17a 37527 jm2.17b 37528 jm2.17c 37529 jm2.24 37530 rmygeid 37531 jm2.18 37555 jm2.19lem1 37556 jm2.19lem2 37557 jm2.19 37560 jm2.22 37562 jm2.23 37563 jm2.20nn 37564 jm2.25 37566 jm2.26a 37567 jm2.26lem3 37568 jm2.26 37569 jm2.15nn0 37570 jm2.16nn0 37571 jm2.27a 37572 jm2.27c 37574 rmydioph 37581 rmxdiophlem 37582 jm3.1lem1 37584 jm3.1 37587 expdiophlem1 37588 |
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