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| Mirrors > Home > MPE Home > Th. List > nfseq | Structured version Visualization version GIF version | ||
| Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfseq.1 | ⊢ Ⅎ𝑥𝑀 |
| nfseq.2 | ⊢ Ⅎ𝑥 + |
| nfseq.3 | ⊢ Ⅎ𝑥𝐹 |
| Ref | Expression |
|---|---|
| nfseq | ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-seq 14043 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥V | |
| 3 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
| 4 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
| 5 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
| 6 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
| 7 | 6, 3 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
| 8 | 4, 5, 7 | nfov 7461 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
| 9 | 3, 8 | nfop 4889 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
| 10 | 2, 2, 9 | nfmpo 7515 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
| 11 | nfseq.1 | . . . . 5 ⊢ Ⅎ𝑥𝑀 | |
| 12 | 6, 11 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
| 13 | 11, 12 | nfop 4889 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
| 14 | 10, 13 | nfrdg 8454 | . . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
| 15 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥ω | |
| 16 | 14, 15 | nfima 6086 | . 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) |
| 17 | 1, 16 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 Vcvv 3480 〈cop 4632 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8449 1c1 11156 + caddc 11158 seqcseq 14042 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 |
| This theorem is referenced by: seqof2 14101 nfsum1 15726 nfsum 15727 nfcprod1 15944 nfcprod 15945 lgamgulm2 27079 binomcxplemdvbinom 44372 binomcxplemdvsum 44374 binomcxplemnotnn0 44375 fmuldfeqlem1 45597 fmuldfeq 45598 sumnnodd 45645 stoweidlem51 46066 |
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