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Mirrors > Home > MPE Home > Th. List > ablpncan2 | Structured version Unicode version |
Description: Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.) |
Ref | Expression |
---|---|
ablsubadd.b |
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ablsubadd.p |
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ablsubadd.m |
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Ref | Expression |
---|---|
ablpncan2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 988 |
. . 3
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2 | simp2 989 |
. . 3
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3 | simp3 990 |
. . 3
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4 | ablsubadd.b |
. . . 4
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5 | ablsubadd.p |
. . . 4
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6 | ablsubadd.m |
. . . 4
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7 | 4, 5, 6 | abladdsub 16428 |
. . 3
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8 | 1, 2, 3, 2, 7 | syl13anc 1221 |
. 2
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9 | ablgrp 16406 |
. . . . 5
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10 | 1, 9 | syl 16 |
. . . 4
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11 | eqid 2454 |
. . . . 5
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12 | 4, 11, 6 | grpsubid 15732 |
. . . 4
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13 | 10, 2, 12 | syl2anc 661 |
. . 3
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14 | 13 | oveq1d 6218 |
. 2
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15 | 4, 5, 11 | grplid 15690 |
. . 3
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16 | 10, 3, 15 | syl2anc 661 |
. 2
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17 | 8, 14, 16 | 3eqtrd 2499 |
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 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-1st 6690 df-2nd 6691 df-0g 14502 df-mnd 15537 df-grp 15667 df-minusg 15668 df-sbg 15669 df-cmn 16403 df-abl 16404 |
This theorem is referenced by: lssvancl1 17152 lspprabs 17302 lsmcv 17348 ttgcontlem1 23303 |
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