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Mirrors > Home > MPE Home > Th. List > addcan | Structured version Visualization version Unicode version |
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcan |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex2 9820 |
. . 3
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2 | 1 | 3ad2ant1 1030 |
. 2
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3 | oveq2 6303 |
. . . 4
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4 | simprr 767 |
. . . . . . 7
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5 | 4 | oveq1d 6310 |
. . . . . 6
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6 | simprl 765 |
. . . . . . 7
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7 | simpl1 1012 |
. . . . . . 7
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8 | simpl2 1013 |
. . . . . . 7
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9 | 6, 7, 8 | addassd 9670 |
. . . . . 6
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10 | addid2 9821 |
. . . . . . 7
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11 | 8, 10 | syl 17 |
. . . . . 6
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12 | 5, 9, 11 | 3eqtr3d 2495 |
. . . . 5
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13 | 4 | oveq1d 6310 |
. . . . . 6
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14 | simpl3 1014 |
. . . . . . 7
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15 | 6, 7, 14 | addassd 9670 |
. . . . . 6
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16 | addid2 9821 |
. . . . . . 7
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17 | 14, 16 | syl 17 |
. . . . . 6
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18 | 13, 15, 17 | 3eqtr3d 2495 |
. . . . 5
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19 | 12, 18 | eqeq12d 2468 |
. . . 4
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20 | 3, 19 | syl5ib 223 |
. . 3
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21 | oveq2 6303 |
. . 3
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22 | 20, 21 | impbid1 207 |
. 2
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23 | 2, 22 | rexlimddv 2885 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-po 4758 df-so 4759 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-pnf 9682 df-mnf 9683 df-ltxr 9685 |
This theorem is referenced by: addcom 9824 addcani 9831 addcomd 9840 addcand 9841 subcan 9934 |
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