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Mirrors > Home > ILE Home > Th. List > seqeq3 | Unicode version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 986 | . . . . . . . 8 | |
2 | 1 | fveq1d 5482 | . . . . . . 7 |
3 | 2 | oveq2d 5852 | . . . . . 6 |
4 | 3 | opeq2d 3759 | . . . . 5 |
5 | 4 | mpoeq3dva 5897 | . . . 4 |
6 | fveq1 5479 | . . . . 5 | |
7 | 6 | opeq2d 3759 | . . . 4 |
8 | freceq1 6351 | . . . . 5 frec frec | |
9 | freceq2 6352 | . . . . 5 frec frec | |
10 | 8, 9 | sylan9eq 2217 | . . . 4 frec frec |
11 | 5, 7, 10 | syl2anc 409 | . . 3 frec frec |
12 | 11 | rneqd 4827 | . 2 frec frec |
13 | df-seqfrec 10371 | . 2 frec | |
14 | df-seqfrec 10371 | . 2 frec | |
15 | 12, 13, 14 | 3eqtr4g 2222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 967 wceq 1342 wcel 2135 cvv 2721 cop 3573 crn 4599 cfv 5182 (class class class)co 5836 cmpo 5838 freccfrec 6349 c1 7745 caddc 7747 cuz 9457 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-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-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-seqfrec 10371 |
This theorem is referenced by: seqeq3d 10378 cbvsum 11287 fsumadd 11333 cbvprod 11485 |
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