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Mirrors > Home > MPE Home > Th. List > receu | Structured version Visualization version Unicode version |
Description: Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.) |
Ref | Expression |
---|---|
receu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recex 10266 |
. . . 4
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2 | 1 | 3adant1 1048 |
. . 3
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3 | simprl 772 |
. . . . . . 7
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4 | simpll 768 |
. . . . . . 7
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5 | 3, 4 | mulcld 9681 |
. . . . . 6
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6 | oveq1 6315 |
. . . . . . . 8
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7 | 6 | ad2antll 743 |
. . . . . . 7
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8 | simplr 770 |
. . . . . . . 8
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9 | 8, 3, 4 | mulassd 9684 |
. . . . . . 7
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10 | 4 | mulid2d 9679 |
. . . . . . 7
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11 | 7, 9, 10 | 3eqtr3d 2513 |
. . . . . 6
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12 | oveq2 6316 |
. . . . . . . 8
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13 | 12 | eqeq1d 2473 |
. . . . . . 7
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14 | 13 | rspcev 3136 |
. . . . . 6
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15 | 5, 11, 14 | syl2anc 673 |
. . . . 5
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16 | 15 | rexlimdvaa 2872 |
. . . 4
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17 | 16 | 3adant3 1050 |
. . 3
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18 | 2, 17 | mpd 15 |
. 2
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19 | eqtr3 2492 |
. . . . . . 7
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20 | mulcan 10271 |
. . . . . . 7
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21 | 19, 20 | syl5ib 227 |
. . . . . 6
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22 | 21 | 3expa 1231 |
. . . . 5
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23 | 22 | expcom 442 |
. . . 4
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24 | 23 | 3adant1 1048 |
. . 3
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25 | 24 | ralrimivv 2813 |
. 2
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26 | oveq2 6316 |
. . . 4
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27 | 26 | eqeq1d 2473 |
. . 3
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28 | 27 | reu4 3220 |
. 2
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29 | 18, 25, 28 | sylanbrc 677 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-po 4760 df-so 4761 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 |
This theorem is referenced by: divmul 10295 divcl 10298 rexdiv 28470 |
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