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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 12664 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
2 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥V | |
3 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
4 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
5 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
6 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
7 | 6, 3 | nffv 6110 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
8 | 4, 5, 7 | nfov 6575 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
9 | 3, 8 | nfop 4356 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
10 | 2, 2, 9 | nfmpt2 6622 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
11 | nfseq.1 | . . . . 5 ⊢ Ⅎ𝑥𝑀 | |
12 | 6, 11 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
13 | 11, 12 | nfop 4356 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
14 | 10, 13 | nfrdg 7397 | . . 3 ⊢ Ⅎ𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
15 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfima 5393 | . 2 ⊢ Ⅎ𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) |
17 | 1, 16 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 Vcvv 3173 〈cop 4131 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 |
This theorem is referenced by: seqof2 12721 nfsum1 14268 nfsum 14269 nfcprod1 14479 nfcprod 14480 lgamgulm2 24562 binomcxplemdvbinom 37574 binomcxplemdvsum 37576 binomcxplemnotnn0 37577 fmuldfeqlem1 38649 fmuldfeq 38650 sumnnodd 38697 stoweidlem51 38944 |
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