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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 13993 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
2 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
4 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
5 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
6 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
7 | 6, 3 | nffv 6901 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
8 | 4, 5, 7 | nfov 7444 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
9 | 3, 8 | nfop 4885 | . . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩ |
10 | 2, 2, 9 | nfmpo 7496 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩) |
11 | nfseq.1 | . . . . 5 ⊢ Ⅎ𝑥𝑀 | |
12 | 6, 11 | nffv 6901 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
13 | 11, 12 | nfop 4885 | . . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩ |
14 | 10, 13 | nfrdg 8428 | . . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) |
15 | nfcv 2899 | . . 3 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfima 6065 | . 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) |
17 | 1, 16 | nfcxfr 2897 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2879 Vcvv 3470 ⟨cop 4630 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ωcom 7864 reccrdg 8423 1c1 11133 + caddc 11135 seqcseq 13992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-xp 5678 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-iota 6494 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-seq 13993 |
This theorem is referenced by: seqof2 14051 nfsum1 15662 nfsum 15663 nfcprod1 15880 nfcprod 15881 lgamgulm2 26961 binomcxplemdvbinom 43784 binomcxplemdvsum 43786 binomcxplemnotnn0 43787 fmuldfeqlem1 44964 fmuldfeq 44965 sumnnodd 45012 stoweidlem51 45433 |
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