![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > seqp1d | Structured version Visualization version GIF version |
Description: Value of the sequence builder function at a successor, deduction form. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by AV, 3-May-2024.) |
Ref | Expression |
---|---|
seqp1d.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
seqp1d.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
seqp1d.3 | ⊢ 𝐾 = (𝑁 + 1) |
seqp1d.4 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴) |
seqp1d.5 | ⊢ (𝜑 → (𝐹‘𝐾) = 𝐵) |
Ref | Expression |
---|---|
seqp1d | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqp1d.3 | . . . 4 ⊢ 𝐾 = (𝑁 + 1) | |
2 | 1 | fveq2i 6849 | . . 3 ⊢ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑁 + 1)) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑁 + 1))) |
4 | seqp1d.2 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
5 | seqp1d.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 4, 5 | eleqtrdi 2844 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | seqp1 13930 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
9 | seqp1d.4 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴) | |
10 | 1 | fveq2i 6849 | . . . 4 ⊢ (𝐹‘𝐾) = (𝐹‘(𝑁 + 1)) |
11 | seqp1d.5 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐾) = 𝐵) | |
12 | 10, 11 | eqtr3id 2787 | . . 3 ⊢ (𝜑 → (𝐹‘(𝑁 + 1)) = 𝐵) |
13 | 9, 12 | oveq12d 7379 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))) = (𝐴 + 𝐵)) |
14 | 3, 8, 13 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6500 (class class class)co 7361 1c1 11060 + caddc 11062 ℤ≥cuz 12771 seqcseq 13915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-seq 13916 |
This theorem is referenced by: seqp1iOLD 13933 climcndslem2 15743 ege2le3 15980 efgt1p2 16004 efgt1p 16005 ovolunlem1a 24883 itcoval1 46839 itcoval2 46840 itcoval3 46841 itcovalsuc 46843 ackvalsuc1mpt 46854 |
Copyright terms: Public domain | W3C validator |