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| Description: Lemma for recex 5749. |
| Ref | Expression |
|---|---|
| recextlem2 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gt0ne0 5683 |
. 2
| |
| 2 | readdcl 5367 |
. . . 4
| |
| 3 | remulcl 5369 |
. . . . 5
| |
| 4 | 3 | anidms 444 |
. . . 4
|
| 5 | remulcl 5369 |
. . . . 5
| |
| 6 | 5 | anidms 444 |
. . . 4
|
| 7 | 2, 4, 6 | syl2an 465 |
. . 3
|
| 8 | 7 | 3adant3 811 |
. 2
|
| 9 | addgtge0 5714 |
. . . . . 6
| |
| 10 | 4, 6 | anim12i 340 |
. . . . . . 7
|
| 11 | 10 | adantr 398 |
. . . . . 6
|
| 12 | msqgt0 5680 |
. . . . . . . 8
| |
| 13 | msqge0 5681 |
. . . . . . . 8
| |
| 14 | 12, 13 | anim12i 340 |
. . . . . . 7
|
| 15 | 14 | an1rs 500 |
. . . . . 6
|
| 16 | 9, 11, 15 | sylanc 482 |
. . . . 5
|
| 17 | addgegt0 5713 |
. . . . . 6
| |
| 18 | 10 | adantr 398 |
. . . . . 6
|
| 19 | msqge0 5681 |
. . . . . . . 8
| |
| 20 | msqgt0 5680 |
. . . . . . . 8
| |
| 21 | 19, 20 | anim12i 340 |
. . . . . . 7
|
| 22 | 21 | anassrs 452 |
. . . . . 6
|
| 23 | 17, 18, 22 | sylanc 482 |
. . . . 5
|
| 24 | 16, 23 | jaodan 435 |
. . . 4
|
| 25 | opreq12 4028 |
. . . . . . . 8
| |
| 26 | opreq2 4027 |
. . . . . . . . 9
| |
| 27 | axicn 5335 |
. . . . . . . . . 10
| |
| 28 | 27 | mul01i 5496 |
. . . . . . . . 9
|
| 29 | 26, 28 | syl6eq 1570 |
. . . . . . . 8
|
| 30 | 25, 29 | sylan2 462 |
. . . . . . 7
|
| 31 | 0cn 5393 |
. . . . . . . 8
| |
| 32 | 31 | addid1i 5395 |
. . . . . . 7
|
| 33 | 30, 32 | syl6eq 1570 |
. . . . . 6
|
| 34 | 33 | necon3ai 1653 |
. . . . 5
|
| 35 | neorian 1687 |
. . . . 5
| |
| 36 | 34, 35 | sylibr 207 |
. . . 4
|
| 37 | 24, 36 | sylan2 462 |
. . 3
|
| 38 | 37 | 3impa 840 |
. 2
|
| 39 | 1, 8, 38 | sylanc 482 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: recex 5749 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1003 ax-gen 1004 ax-8 1005 ax-9 1006 ax-10 1007 ax-11 1008 ax-12 1009 ax-13 1010 ax-14 1011 ax-17 1012 ax-4 1014 ax-5o 1016 ax-6o 1019 ax-9o 1164 ax-10o 1182 ax-16 1252 ax-11o 1260 ax-ext 1504 ax-rep 2748 ax-sep 2758 ax-nul 2765 ax-pow 2798 ax-pr 2835 ax-un 2922 ax-inf2 4687 |
| This theorem depends on definitions: df-bi 154 df-or 231 df-an 232 df-3or 788 df-3an 789 df-ex 1022 df-sb 1214 df-eu 1424 df-mo 1425 df-clab 1510 df-cleq 1515 df-clel 1518 df-ne 1634 df-nel 1635 df-ral 1696 df-rex 1697 df-reu 1698 df-rab 1699 df-v 1859 df-sbc 1989 df-csb 2052 df-dif 2100 df-un 2101 df-in 2102 df-ss 2104 df-pss 2106 df-nul 2332 df-if 2414 df-pw 2454 df-sn 2464 df-pr 2465 df-tp 2467 df-op 2468 df-uni 2558 df-int 2588 df-iun 2622 df-br 2675 df-opab 2722 df-tr 2736 df-eprel 2888 df-id 2891 df-po 2896 df-so 2906 df-fr 2974 df-we 2991 df-ord 3008 df-on 3009 df-lim 3010 df-suc 3011 df-om 3189 df-xp 3241 df-rel 3242 df-cnv 3243 df-co 3244 df-dm 3245 df-rn 3246 df-res 3247 df-ima 3248 df-fun 3249 df-fn 3250 df-f 3251 df-f1 3252 df-fo 3253 df-f1o 3254 df-fv 3255 df-rdg 3990 df-opr 4023 df-oprab 4024 df-1st 4137 df-2nd 4138 df-1o 4191 df-oadd 4193 df-omul 4194 df-er 4319 df-ec 4321 df-qs 4324 df-en 4429 df-dom 4430 df-sdom 4431 df-ni 5065 df-pli 5066 df-mi 5067 df-lti 5068 df-plpq 5100 df-mpq 5101 df-enq 5102 df-nq 5103 df-plq 5104 df-mq 5105 df-rq 5106 df-ltq 5107 df-1q 5108 df-np 5151 df-1p 5152 df-plp 5153 df-mp 5154 df-ltp 5155 df-plpr 5229 df-mpr 5230 df-enr 5231 df-nr 5232 df-plr 5233 df-mr 5234 df-ltr 5235 df-0r 5236 df-1r 5237 df-m1r 5238 df-c 5305 df-0 5306 df-1 5307 df-i 5308 df-r 5309 df-plus 5310 df-mul 5311 df-lt 5312 df-sub 5421 df-neg 5423 df-pnf 5552 df-mnf 5553 df-xr 5554 df-ltxr 5555 df-le 5556 |