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Mirrors > Home > ILE Home > Th. List > seq3caopr3 | Unicode version |
Description: Lemma for seq3caopr2 10407. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.) |
Ref | Expression |
---|---|
seqcaopr3.1 | |
seqcaopr3.2 | |
seqcaopr3.3 | |
seq3caopr3.4 | |
seq3caopr3.5 | |
seq3caopr3.6 | |
seqcaopr3.7 | ..^ |
Ref | Expression |
---|---|
seq3caopr3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr3.3 | . . 3 | |
2 | eluzfz2 9957 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveq2 5480 | . . . . 5 | |
5 | fveq2 5480 | . . . . . 6 | |
6 | fveq2 5480 | . . . . . 6 | |
7 | 5, 6 | oveq12d 5854 | . . . . 5 |
8 | 4, 7 | eqeq12d 2179 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | fveq2 5480 | . . . . 5 | |
11 | fveq2 5480 | . . . . . 6 | |
12 | fveq2 5480 | . . . . . 6 | |
13 | 11, 12 | oveq12d 5854 | . . . . 5 |
14 | 10, 13 | eqeq12d 2179 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | fveq2 5480 | . . . . 5 | |
17 | fveq2 5480 | . . . . . 6 | |
18 | fveq2 5480 | . . . . . 6 | |
19 | 17, 18 | oveq12d 5854 | . . . . 5 |
20 | 16, 19 | eqeq12d 2179 | . . . 4 |
21 | 20 | imbi2d 229 | . . 3 |
22 | fveq2 5480 | . . . . 5 | |
23 | fveq2 5480 | . . . . . 6 | |
24 | fveq2 5480 | . . . . . 6 | |
25 | 23, 24 | oveq12d 5854 | . . . . 5 |
26 | 22, 25 | eqeq12d 2179 | . . . 4 |
27 | 26 | imbi2d 229 | . . 3 |
28 | fveq2 5480 | . . . . . . 7 | |
29 | fveq2 5480 | . . . . . . . 8 | |
30 | fveq2 5480 | . . . . . . . 8 | |
31 | 29, 30 | oveq12d 5854 | . . . . . . 7 |
32 | 28, 31 | eqeq12d 2179 | . . . . . 6 |
33 | seq3caopr3.6 | . . . . . . 7 | |
34 | 33 | ralrimiva 2537 | . . . . . 6 |
35 | eluzel2 9462 | . . . . . . . 8 | |
36 | 1, 35 | syl 14 | . . . . . . 7 |
37 | uzid 9471 | . . . . . . 7 | |
38 | 36, 37 | syl 14 | . . . . . 6 |
39 | 32, 34, 38 | rspcdva 2830 | . . . . 5 |
40 | seqcaopr3.2 | . . . . . . . . . . . 12 | |
41 | 40 | ralrimivva 2546 | . . . . . . . . . . 11 |
42 | 41 | adantr 274 | . . . . . . . . . 10 |
43 | seq3caopr3.4 | . . . . . . . . . . 11 | |
44 | seq3caopr3.5 | . . . . . . . . . . 11 | |
45 | oveq1 5843 | . . . . . . . . . . . . 13 | |
46 | 45 | eleq1d 2233 | . . . . . . . . . . . 12 |
47 | oveq2 5844 | . . . . . . . . . . . . 13 | |
48 | 47 | eleq1d 2233 | . . . . . . . . . . . 12 |
49 | 46, 48 | rspc2v 2838 | . . . . . . . . . . 11 |
50 | 43, 44, 49 | syl2anc 409 | . . . . . . . . . 10 |
51 | 42, 50 | mpd 13 | . . . . . . . . 9 |
52 | 33, 51 | eqeltrd 2241 | . . . . . . . 8 |
53 | 52 | ralrimiva 2537 | . . . . . . 7 |
54 | fveq2 5480 | . . . . . . . . 9 | |
55 | 54 | eleq1d 2233 | . . . . . . . 8 |
56 | 55 | rspcv 2821 | . . . . . . 7 |
57 | 53, 56 | mpan9 279 | . . . . . 6 |
58 | seqcaopr3.1 | . . . . . 6 | |
59 | 36, 57, 58 | seq3-1 10385 | . . . . 5 |
60 | 43 | ralrimiva 2537 | . . . . . . . 8 |
61 | fveq2 5480 | . . . . . . . . . 10 | |
62 | 61 | eleq1d 2233 | . . . . . . . . 9 |
63 | 62 | rspcv 2821 | . . . . . . . 8 |
64 | 60, 63 | mpan9 279 | . . . . . . 7 |
65 | 36, 64, 58 | seq3-1 10385 | . . . . . 6 |
66 | 44 | ralrimiva 2537 | . . . . . . . 8 |
67 | fveq2 5480 | . . . . . . . . . 10 | |
68 | 67 | eleq1d 2233 | . . . . . . . . 9 |
69 | 68 | rspcv 2821 | . . . . . . . 8 |
70 | 66, 69 | mpan9 279 | . . . . . . 7 |
71 | 36, 70, 58 | seq3-1 10385 | . . . . . 6 |
72 | 65, 71 | oveq12d 5854 | . . . . 5 |
73 | 39, 59, 72 | 3eqtr4d 2207 | . . . 4 |
74 | 73 | a1i 9 | . . 3 |
75 | oveq1 5843 | . . . . . 6 | |
76 | elfzouz 10076 | . . . . . . . . 9 ..^ | |
77 | 76 | adantl 275 | . . . . . . . 8 ..^ |
78 | 57 | adantlr 469 | . . . . . . . 8 ..^ |
79 | 58 | adantlr 469 | . . . . . . . 8 ..^ |
80 | 77, 78, 79 | seq3p1 10387 | . . . . . . 7 ..^ |
81 | seqcaopr3.7 | . . . . . . . 8 ..^ | |
82 | fveq2 5480 | . . . . . . . . . . 11 | |
83 | fveq2 5480 | . . . . . . . . . . . 12 | |
84 | fveq2 5480 | . . . . . . . . . . . 12 | |
85 | 83, 84 | oveq12d 5854 | . . . . . . . . . . 11 |
86 | 82, 85 | eqeq12d 2179 | . . . . . . . . . 10 |
87 | 34 | adantr 274 | . . . . . . . . . 10 ..^ |
88 | fzofzp1 10152 | . . . . . . . . . . . 12 ..^ | |
89 | elfzuz 9947 | . . . . . . . . . . . 12 | |
90 | 88, 89 | syl 14 | . . . . . . . . . . 11 ..^ |
91 | 90 | adantl 275 | . . . . . . . . . 10 ..^ |
92 | 86, 87, 91 | rspcdva 2830 | . . . . . . . . 9 ..^ |
93 | 92 | oveq2d 5852 | . . . . . . . 8 ..^ |
94 | 64 | adantlr 469 | . . . . . . . . . 10 ..^ |
95 | 77, 94, 79 | seq3p1 10387 | . . . . . . . . 9 ..^ |
96 | 70 | adantlr 469 | . . . . . . . . . 10 ..^ |
97 | 77, 96, 79 | seq3p1 10387 | . . . . . . . . 9 ..^ |
98 | 95, 97 | oveq12d 5854 | . . . . . . . 8 ..^ |
99 | 81, 93, 98 | 3eqtr4rd 2208 | . . . . . . 7 ..^ |
100 | 80, 99 | eqeq12d 2179 | . . . . . 6 ..^ |
101 | 75, 100 | syl5ibr 155 | . . . . 5 ..^ |
102 | 101 | expcom 115 | . . . 4 ..^ |
103 | 102 | a2d 26 | . . 3 ..^ |
104 | 9, 15, 21, 27, 74, 103 | fzind2 10164 | . 2 |
105 | 3, 104 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wral 2442 cfv 5182 (class class class)co 5836 c1 7745 caddc 7747 cz 9182 cuz 9457 cfz 9935 ..^cfzo 10067 cseq 10370 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-fzo 10068 df-seqfrec 10371 |
This theorem is referenced by: seq3caopr2 10407 |
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