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Mirrors > Home > ILE Home > Th. List > sumrbdclem | Unicode version |
Description: Lemma for sumrbdc 11329. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.) |
Ref | Expression |
---|---|
isummo.1 | |
isummo.2 | |
isummo.dc | DECID |
isumrb.3 |
Ref | Expression |
---|---|
sumrbdclem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addid2 8045 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | 0cnd 7900 | . 2 | |
4 | isumrb.3 | . . 3 | |
5 | 4 | adantr 274 | . 2 |
6 | eluzelz 9483 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | isummo.dc | . . . . . . . . 9 DECID | |
9 | exmiddc 831 | . . . . . . . . 9 DECID | |
10 | 8, 9 | syl 14 | . . . . . . . 8 |
11 | iftrue 3530 | . . . . . . . . . . . . 13 | |
12 | 11 | adantl 275 | . . . . . . . . . . . 12 |
13 | isummo.2 | . . . . . . . . . . . 12 | |
14 | 12, 13 | eqeltrd 2247 | . . . . . . . . . . 11 |
15 | 14 | ex 114 | . . . . . . . . . 10 |
16 | iffalse 3533 | . . . . . . . . . . . 12 | |
17 | 0cn 7899 | . . . . . . . . . . . 12 | |
18 | 16, 17 | eqeltrdi 2261 | . . . . . . . . . . 11 |
19 | 18 | a1i 9 | . . . . . . . . . 10 |
20 | 15, 19 | jaod 712 | . . . . . . . . 9 |
21 | 20 | adantr 274 | . . . . . . . 8 |
22 | 10, 21 | mpd 13 | . . . . . . 7 |
23 | 22 | ralrimiva 2543 | . . . . . 6 |
24 | nfv 1521 | . . . . . . . . 9 | |
25 | nfcsb1v 3082 | . . . . . . . . 9 | |
26 | nfcv 2312 | . . . . . . . . 9 | |
27 | 24, 25, 26 | nfif 3553 | . . . . . . . 8 |
28 | 27 | nfel1 2323 | . . . . . . 7 |
29 | eleq1 2233 | . . . . . . . . 9 | |
30 | csbeq1a 3058 | . . . . . . . . 9 | |
31 | 29, 30 | ifbieq1d 3547 | . . . . . . . 8 |
32 | 31 | eleq1d 2239 | . . . . . . 7 |
33 | 28, 32 | rspc 2828 | . . . . . 6 |
34 | 4, 23, 33 | sylc 62 | . . . . 5 |
35 | 34 | adantr 274 | . . . 4 |
36 | nfcv 2312 | . . . . 5 | |
37 | isummo.1 | . . . . 5 | |
38 | 36, 27, 31, 37 | fvmptf 5586 | . . . 4 |
39 | 7, 35, 38 | syl2anc 409 | . . 3 |
40 | 39, 35 | eqeltrd 2247 | . 2 |
41 | elfzelz 9968 | . . . 4 | |
42 | elfzuz 9964 | . . . . . 6 | |
43 | 42 | adantl 275 | . . . . 5 |
44 | 23 | ad2antrr 485 | . . . . 5 |
45 | nfv 1521 | . . . . . . . 8 | |
46 | nfcsb1v 3082 | . . . . . . . 8 | |
47 | 45, 46, 26 | nfif 3553 | . . . . . . 7 |
48 | 47 | nfel1 2323 | . . . . . 6 |
49 | eleq1 2233 | . . . . . . . 8 | |
50 | csbeq1a 3058 | . . . . . . . 8 | |
51 | 49, 50 | ifbieq1d 3547 | . . . . . . 7 |
52 | 51 | eleq1d 2239 | . . . . . 6 |
53 | 48, 52 | rspc 2828 | . . . . 5 |
54 | 43, 44, 53 | sylc 62 | . . . 4 |
55 | nfcv 2312 | . . . . 5 | |
56 | 55, 47, 51, 37 | fvmptf 5586 | . . . 4 |
57 | 41, 54, 56 | syl2an2 589 | . . 3 |
58 | uznfz 10046 | . . . . . . 7 | |
59 | 58 | con2i 622 | . . . . . 6 |
60 | 59 | adantl 275 | . . . . 5 |
61 | ssel 3141 | . . . . . 6 | |
62 | 61 | ad2antlr 486 | . . . . 5 |
63 | 60, 62 | mtod 658 | . . . 4 |
64 | 63 | iffalsed 3535 | . . 3 |
65 | 57, 64 | eqtrd 2203 | . 2 |
66 | eluzelz 9483 | . . . 4 | |
67 | simpr 109 | . . . . 5 | |
68 | 23 | ad2antrr 485 | . . . . 5 |
69 | 67, 68, 53 | sylc 62 | . . . 4 |
70 | 66, 69, 56 | syl2an2 589 | . . 3 |
71 | 70, 69 | eqeltrd 2247 | . 2 |
72 | addcl 7886 | . . 3 | |
73 | 72 | adantl 275 | . 2 |
74 | 2, 3, 5, 40, 65, 71, 73 | seq3id 10451 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 csb 3049 wss 3121 cif 3525 cmpt 4048 cres 4611 cfv 5196 (class class class)co 5850 cc 7759 cc0 7761 c1 7762 caddc 7764 cmin 8077 cz 9199 cuz 9474 cfz 9952 cseq 10388 |
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-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 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-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 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-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-fz 9953 df-fzo 10086 df-seqfrec 10389 |
This theorem is referenced by: sumrbdc 11329 |
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