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Mirrors > Home > MPE Home > Th. List > lemulge11 | Structured version Unicode version |
Description: Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
Ref | Expression |
---|---|
lemulge11 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1rid 9453 |
. . 3
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2 | 1 | ad2antrr 725 |
. 2
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3 | simpll 753 |
. . . . 5
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4 | simprl 755 |
. . . . 5
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5 | 3, 4 | jca 532 |
. . . 4
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6 | simplr 754 |
. . . . 5
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7 | 1re 9486 |
. . . . . 6
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8 | 0le1 9964 |
. . . . . 6
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9 | 7, 8 | pm3.2i 455 |
. . . . 5
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10 | 6, 9 | jctil 537 |
. . . 4
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11 | 5, 3, 10 | jca31 534 |
. . 3
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12 | leid 9571 |
. . . . 5
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13 | 12 | ad2antrr 725 |
. . . 4
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14 | simprr 756 |
. . . 4
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15 | 13, 14 | jca 532 |
. . 3
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16 | lemul12a 10288 |
. . 3
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17 | 11, 15, 16 | sylc 60 |
. 2
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18 | 2, 17 | eqbrtrrd 4412 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 ax-resscn 9440 ax-1cn 9441 ax-icn 9442 ax-addcl 9443 ax-addrcl 9444 ax-mulcl 9445 ax-mulrcl 9446 ax-mulcom 9447 ax-addass 9448 ax-mulass 9449 ax-distr 9450 ax-i2m1 9451 ax-1ne0 9452 ax-1rid 9453 ax-rnegex 9454 ax-rrecex 9455 ax-cnre 9456 ax-pre-lttri 9457 ax-pre-lttrn 9458 ax-pre-ltadd 9459 ax-pre-mulgt0 9460 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-po 4739 df-so 4740 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-riota 6151 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-er 7201 df-en 7411 df-dom 7412 df-sdom 7413 df-pnf 9521 df-mnf 9522 df-xr 9523 df-ltxr 9524 df-le 9525 df-sub 9698 df-neg 9699 |
This theorem is referenced by: lemulge12 10293 lemulge11d 10371 faclbnd 12167 divalglem1 13700 |
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