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Mirrors > Home > ILE Home > Th. List > axpre-mulgt0 | Unicode version |
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 6800. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-mulgt0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 6727 |
. 2
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2 | elreal 6727 |
. 2
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3 | breq2 3759 |
. . . 4
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4 | 3 | anbi1d 438 |
. . 3
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5 | oveq1 5462 |
. . . 4
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6 | 5 | breq2d 3767 |
. . 3
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7 | 4, 6 | imbi12d 223 |
. 2
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8 | breq2 3759 |
. . . 4
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9 | 8 | anbi2d 437 |
. . 3
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10 | oveq2 5463 |
. . . 4
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11 | 10 | breq2d 3767 |
. . 3
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12 | 9, 11 | imbi12d 223 |
. 2
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13 | df-0 6718 |
. . . . . 6
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14 | 13 | breq1i 3762 |
. . . . 5
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15 | ltresr 6736 |
. . . . 5
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16 | 14, 15 | bitri 173 |
. . . 4
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17 | 13 | breq1i 3762 |
. . . . 5
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18 | ltresr 6736 |
. . . . 5
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19 | 17, 18 | bitri 173 |
. . . 4
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20 | mulgt0sr 6704 |
. . . 4
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21 | 16, 19, 20 | syl2anb 275 |
. . 3
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22 | 13 | a1i 9 |
. . . . 5
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23 | mulresr 6735 |
. . . . 5
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24 | 22, 23 | breq12d 3768 |
. . . 4
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25 | ltresr 6736 |
. . . 4
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26 | 24, 25 | syl6bb 185 |
. . 3
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27 | 21, 26 | syl5ibr 145 |
. 2
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28 | 1, 2, 7, 12, 27 | 2gencl 2581 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-imp 6452 df-iltp 6453 df-enr 6654 df-nr 6655 df-plr 6656 df-mr 6657 df-ltr 6658 df-0r 6659 df-m1r 6661 df-c 6717 df-0 6718 df-r 6721 df-mul 6723 df-lt 6724 |
This theorem is referenced by: (None) |
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