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Mirrors > Home > MPE Home > Th. List > rhmmul | Structured version Visualization version Unicode version |
Description: A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
rhmmul.x |
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rhmmul.m |
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rhmmul.n |
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Ref | Expression |
---|---|
rhmmul |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2451 |
. . 3
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2 | eqid 2451 |
. . 3
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3 | 1, 2 | rhmmhm 17950 |
. 2
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4 | rhmmul.x |
. . . 4
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5 | 1, 4 | mgpbas 17729 |
. . 3
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6 | rhmmul.m |
. . . 4
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7 | 1, 6 | mgpplusg 17727 |
. . 3
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8 | rhmmul.n |
. . . 4
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9 | 2, 8 | mgpplusg 17727 |
. . 3
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10 | 5, 7, 9 | mhmlin 16589 |
. 2
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11 | 3, 10 | syl3an1 1301 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-plusg 15203 df-0g 15340 df-mhm 16582 df-ghm 16881 df-mgp 17724 df-ur 17736 df-ring 17782 df-rnghom 17943 |
This theorem is referenced by: srngmul 18086 evl1muld 18931 evl1scvarpw 18951 domnchr 19103 znfld 19131 znidomb 19132 znunit 19134 znrrg 19136 mat2pmatmul 19755 mat2pmatlin 19759 cayhamlem4 19912 ply1rem 23114 fta1glem2 23117 fta1blem 23119 dchrzrhmul 24174 lgsdchr 24276 lgseisenlem3 24279 lgseisenlem4 24280 rhmdvdsr 28581 rhmopp 28582 rhmdvd 28584 rhmunitinv 28585 kerunit 28586 mdetpmtr1 28649 mdetpmtr12 28651 qqhghm 28792 qqhrhm 28793 |
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