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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 13365 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
2 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
4 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
5 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
6 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
7 | 6, 3 | nffv 6655 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
8 | 4, 5, 7 | nfov 7165 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
9 | 3, 8 | nfop 4781 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
10 | 2, 2, 9 | nfmpo 7215 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
11 | nfseq.1 | . . . . 5 ⊢ Ⅎ𝑥𝑀 | |
12 | 6, 11 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
13 | 11, 12 | nfop 4781 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
14 | 10, 13 | nfrdg 8033 | . . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
15 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfima 5904 | . 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) |
17 | 1, 16 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2936 Vcvv 3441 〈cop 4531 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ωcom 7560 reccrdg 8028 1c1 10527 + caddc 10529 seqcseq 13364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seq 13365 |
This theorem is referenced by: seqof2 13424 nfsum1 15038 nfsum 15039 nfsumOLD 15040 nfcprod1 15256 nfcprod 15257 lgamgulm2 25621 binomcxplemdvbinom 41057 binomcxplemdvsum 41059 binomcxplemnotnn0 41060 fmuldfeqlem1 42224 fmuldfeq 42225 sumnnodd 42272 stoweidlem51 42693 |
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