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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 13968 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) | |
2 | nfcv 2895 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2895 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
4 | nfcv 2895 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
5 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
6 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
7 | 6, 3 | nffv 6892 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
8 | 4, 5, 7 | nfov 7432 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
9 | 3, 8 | nfop 4882 | . . . . 5 ⊢ Ⅎ𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩ |
10 | 2, 2, 9 | nfmpo 7484 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩) |
11 | nfseq.1 | . . . . 5 ⊢ Ⅎ𝑥𝑀 | |
12 | 6, 11 | nffv 6892 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
13 | 11, 12 | nfop 4882 | . . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩ |
14 | 10, 13 | nfrdg 8410 | . . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) |
15 | nfcv 2895 | . . 3 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfima 6058 | . 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) |
17 | 1, 16 | nfcxfr 2893 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2875 Vcvv 3466 ⟨cop 4627 “ cima 5670 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 ωcom 7849 reccrdg 8405 1c1 11108 + caddc 11110 seqcseq 13967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-iota 6486 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-seq 13968 |
This theorem is referenced by: seqof2 14027 nfsum1 15638 nfsum 15639 nfcprod1 15856 nfcprod 15857 lgamgulm2 26908 binomcxplemdvbinom 43661 binomcxplemdvsum 43663 binomcxplemnotnn0 43664 fmuldfeqlem1 44843 fmuldfeq 44844 sumnnodd 44891 stoweidlem51 45312 |
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