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Mirrors > Home > MPE Home > Th. List > msq11 | Structured version Visualization version Unicode version |
Description: The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
msq11 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | le2msq 10495 |
. . 3
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2 | le2msq 10495 |
. . . 4
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3 | 2 | ancoms 459 |
. . 3
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4 | 1, 3 | anbi12d 722 |
. 2
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5 | simpll 765 |
. . 3
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6 | simprl 769 |
. . 3
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7 | 5, 6 | letri3d 9764 |
. 2
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8 | 5, 5 | remulcld 9658 |
. . 3
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9 | 6, 6 | remulcld 9658 |
. . 3
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10 | 8, 9 | letri3d 9764 |
. 2
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11 | 4, 7, 10 | 3bitr4rd 294 |
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 1673 ax-4 1686 ax-5 1762 ax-6 1809 ax-7 1855 ax-8 1893 ax-9 1900 ax-10 1919 ax-11 1924 ax-12 1937 ax-13 2092 ax-ext 2432 ax-sep 4497 ax-nul 4506 ax-pow 4554 ax-pr 4612 ax-un 6571 ax-resscn 9583 ax-1cn 9584 ax-icn 9585 ax-addcl 9586 ax-addrcl 9587 ax-mulcl 9588 ax-mulrcl 9589 ax-mulcom 9590 ax-addass 9591 ax-mulass 9592 ax-distr 9593 ax-i2m1 9594 ax-1ne0 9595 ax-1rid 9596 ax-rnegex 9597 ax-rrecex 9598 ax-cnre 9599 ax-pre-lttri 9600 ax-pre-lttrn 9601 ax-pre-ltadd 9602 ax-pre-mulgt0 9603 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1451 df-ex 1668 df-nf 1672 df-sb 1802 df-eu 2304 df-mo 2305 df-clab 2439 df-cleq 2445 df-clel 2448 df-nfc 2582 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3015 df-sbc 3236 df-csb 3332 df-dif 3375 df-un 3377 df-in 3379 df-ss 3386 df-nul 3700 df-if 3850 df-pw 3921 df-sn 3937 df-pr 3939 df-op 3943 df-uni 4169 df-br 4375 df-opab 4434 df-mpt 4435 df-id 4727 df-po 4733 df-so 4734 df-xp 4818 df-rel 4819 df-cnv 4820 df-co 4821 df-dm 4822 df-rn 4823 df-res 4824 df-ima 4825 df-iota 5525 df-fun 5563 df-fn 5564 df-f 5565 df-f1 5566 df-fo 5567 df-f1o 5568 df-fv 5569 df-riota 6238 df-ov 6279 df-oprab 6280 df-mpt2 6281 df-er 7350 df-en 7557 df-dom 7558 df-sdom 7559 df-pnf 9664 df-mnf 9665 df-xr 9666 df-ltxr 9667 df-le 9668 df-sub 9849 df-neg 9850 |
This theorem is referenced by: msq11i 10510 sq11 12341 |
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