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Mirrors > Home > ILE Home > Th. List > recexaplem2 | Unicode version |
Description: Lemma for recexap 8558. (Contributed by Jim Kingdon, 20-Feb-2020.) |
Ref | Expression |
---|---|
recexaplem2 | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7856 | . . . . . . . . . . 11 | |
2 | 1 | mul01i 8297 | . . . . . . . . . 10 |
3 | 2 | oveq2i 5861 | . . . . . . . . 9 |
4 | 00id 8047 | . . . . . . . . 9 | |
5 | 3, 4 | eqtr2i 2192 | . . . . . . . 8 |
6 | 5 | breq2i 3995 | . . . . . . 7 # # |
7 | 0re 7907 | . . . . . . . 8 | |
8 | apreim 8509 | . . . . . . . 8 # # # | |
9 | 7, 7, 8 | mpanr12 437 | . . . . . . 7 # # # |
10 | 6, 9 | syl5bb 191 | . . . . . 6 # # # |
11 | 10 | pm5.32i 451 | . . . . 5 # # # |
12 | remulcl 7889 | . . . . . . . . . 10 | |
13 | 12 | anidms 395 | . . . . . . . . 9 |
14 | remulcl 7889 | . . . . . . . . . 10 | |
15 | 14 | anidms 395 | . . . . . . . . 9 |
16 | 13, 15 | anim12i 336 | . . . . . . . 8 |
17 | 16 | adantr 274 | . . . . . . 7 # |
18 | apsqgt0 8507 | . . . . . . . . 9 # | |
19 | msqge0 8522 | . . . . . . . . 9 | |
20 | 18, 19 | anim12i 336 | . . . . . . . 8 # |
21 | 20 | an32s 563 | . . . . . . 7 # |
22 | addgtge0 8356 | . . . . . . 7 | |
23 | 17, 21, 22 | syl2anc 409 | . . . . . 6 # |
24 | 16 | adantr 274 | . . . . . . 7 # |
25 | msqge0 8522 | . . . . . . . . 9 | |
26 | apsqgt0 8507 | . . . . . . . . 9 # | |
27 | 25, 26 | anim12i 336 | . . . . . . . 8 # |
28 | 27 | anassrs 398 | . . . . . . 7 # |
29 | addgegt0 8355 | . . . . . . 7 | |
30 | 24, 28, 29 | syl2anc 409 | . . . . . 6 # |
31 | 23, 30 | jaodan 792 | . . . . 5 # # |
32 | 11, 31 | sylbi 120 | . . . 4 # |
33 | 32 | 3impa 1189 | . . 3 # |
34 | 33 | olcd 729 | . 2 # |
35 | simp1 992 | . . . . 5 # | |
36 | 35, 35 | remulcld 7937 | . . . 4 # |
37 | simp2 993 | . . . . 5 # | |
38 | 37, 37 | remulcld 7937 | . . . 4 # |
39 | 36, 38 | readdcld 7936 | . . 3 # |
40 | reaplt 8494 | . . 3 # | |
41 | 39, 7, 40 | sylancl 411 | . 2 # # |
42 | 34, 41 | mpbird 166 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 w3a 973 wcel 2141 class class class wbr 3987 (class class class)co 5850 cr 7760 cc0 7761 ci 7763 caddc 7764 cmul 7766 clt 7941 cle 7942 # cap 8487 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 |
This theorem is referenced by: recexap 8558 |
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