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Mirrors > Home > ILE Home > Th. List > msqge0 | Unicode version |
Description: A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
msqge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | remulcl 7902 | . . . . 5 | |
2 | 1 | anidms 395 | . . . 4 |
3 | 0re 7920 | . . . 4 | |
4 | ltnsym2 8010 | . . . 4 | |
5 | 2, 3, 4 | sylancl 411 | . . 3 |
6 | orc 707 | . . . . . 6 | |
7 | reaplt 8507 | . . . . . . 7 # | |
8 | 2, 3, 7 | sylancl 411 | . . . . . 6 # |
9 | 6, 8 | syl5ibr 155 | . . . . 5 # |
10 | recn 7907 | . . . . . . . . 9 | |
11 | mulap0r 8534 | . . . . . . . . . 10 # # # | |
12 | 10, 11 | syl3an1 1266 | . . . . . . . . 9 # # # |
13 | 10, 12 | syl3an2 1267 | . . . . . . . 8 # # # |
14 | 13 | simpld 111 | . . . . . . 7 # # |
15 | 14 | 3expia 1200 | . . . . . 6 # # |
16 | 15 | anidms 395 | . . . . 5 # # |
17 | apsqgt0 8520 | . . . . . 6 # | |
18 | 17 | ex 114 | . . . . 5 # |
19 | 9, 16, 18 | 3syld 57 | . . . 4 |
20 | 19 | ancld 323 | . . 3 |
21 | 5, 20 | mtod 658 | . 2 |
22 | lenlt 7995 | . . 3 | |
23 | 3, 2, 22 | sylancr 412 | . 2 |
24 | 21, 23 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wcel 2141 class class class wbr 3989 (class class class)co 5853 cc 7772 cr 7773 cc0 7774 cmul 7779 clt 7954 cle 7955 # cap 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 |
This theorem is referenced by: msqge0i 8536 msqge0d 8537 recexaplem2 8570 sqge0 10552 bernneq 10596 |
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