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Mirrors > Home > ILE Home > Th. List > iseqf1olemfvp | Unicode version |
Description: Lemma for seq3f1o 10588. (Contributed by Jim Kingdon, 30-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemfvp.k |
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iseqf1olemfvp.t |
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iseqf1olemfvp.a |
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iseqf1olemfvp.g |
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iseqf1olemfvp.p |
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Ref | Expression |
---|---|
iseqf1olemfvp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemfvp.p |
. . . . 5
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2 | 1 | csbeq2i 3107 |
. . . 4
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3 | iseqf1olemfvp.t |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | f1of 5500 |
. . . . . . 7
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5 | 3, 4 | syl 14 |
. . . . . 6
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6 | iseqf1olemfvp.k |
. . . . . . . 8
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7 | elfzel1 10090 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | syl 14 |
. . . . . . 7
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9 | elfzel2 10089 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 6, 9 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 8, 10 | fzfigd 10502 |
. . . . . 6
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12 | fex 5787 |
. . . . . 6
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13 | 5, 11, 12 | syl2anc 411 |
. . . . 5
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14 | nfcvd 2337 |
. . . . . 6
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15 | fveq1 5553 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | fveq2d 5558 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | ifeq1d 3574 |
. . . . . . 7
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18 | 17 | mpteq2dv 4120 |
. . . . . 6
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19 | 14, 18 | csbiegf 3124 |
. . . . 5
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20 | 13, 19 | syl 14 |
. . . 4
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21 | 2, 20 | eqtrid 2238 |
. . 3
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22 | simpr 110 |
. . . . 5
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23 | 22 | breq1d 4039 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 22 | fveq2d 5558 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | fveq2d 5558 |
. . . 4
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26 | 23, 25 | ifbieq1d 3579 |
. . 3
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27 | iseqf1olemfvp.a |
. . . 4
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28 | elfzuz 10087 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 27, 28 | syl 14 |
. . 3
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30 | elfzle2 10094 |
. . . . . 6
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31 | 27, 30 | syl 14 |
. . . . 5
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32 | 31 | iftrued 3564 |
. . . 4
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33 | fveq2 5554 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 33 | eleq1d 2262 |
. . . . 5
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35 | iseqf1olemfvp.g |
. . . . . 6
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36 | 35 | ralrimiva 2567 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 5, 27 | ffvelcdmd 5694 |
. . . . . 6
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38 | elfzuz 10087 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 37, 38 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34, 36, 39 | rspcdva 2869 |
. . . 4
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41 | 32, 40 | eqeltrd 2270 |
. . 3
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42 | 21, 26, 29, 41 | fvmptd 5638 |
. 2
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43 | 42, 32 | eqtrd 2226 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-er 6587 df-en 6795 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-fz 10075 |
This theorem is referenced by: seq3f1olemqsumkj 10582 seq3f1olemqsumk 10583 |
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