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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmncan2 | Structured version Unicode version |
Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
dmncan.1 |
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dmncan.2 |
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dmncan.3 |
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dmncan.4 |
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Ref | Expression |
---|---|
dmncan2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmncrng 28997 |
. . . 4
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2 | dmncan.1 |
. . . . . . 7
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3 | dmncan.2 |
. . . . . . 7
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4 | dmncan.3 |
. . . . . . 7
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5 | 2, 3, 4 | crngocom 28942 |
. . . . . 6
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6 | 5 | 3adant3r2 1198 |
. . . . 5
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7 | 2, 3, 4 | crngocom 28942 |
. . . . . 6
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8 | 7 | 3adant3r1 1197 |
. . . . 5
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9 | 6, 8 | eqeq12d 2473 |
. . . 4
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10 | 1, 9 | sylan 471 |
. . 3
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11 | 10 | adantr 465 |
. 2
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12 | 3anrot 970 |
. . . 4
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13 | 12 | biimpri 206 |
. . 3
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14 | dmncan.4 |
. . . 4
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15 | 2, 3, 4, 14 | dmncan1 29017 |
. . 3
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16 | 13, 15 | sylanl2 651 |
. 2
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17 | 11, 16 | sylbid 215 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-1st 6680 df-2nd 6681 df-1o 7023 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-grpo 23823 df-gid 23824 df-ginv 23825 df-gdiv 23826 df-ablo 23914 df-ass 23945 df-exid 23947 df-mgm 23951 df-sgr 23963 df-mndo 23970 df-rngo 24008 df-com2 24043 df-crngo 28937 df-idl 28951 df-pridl 28952 df-prrngo 28989 df-dmn 28990 df-igen 29001 |
This theorem is referenced by: (None) |
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