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Mirrors > Home > MPE Home > Th. List > seqm1 | Structured version Visualization version GIF version |
Description: Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) |
Ref | Expression |
---|---|
seqm1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzp1m1 12872 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) | |
2 | seqp1 14007 | . . 3 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘((𝑁 − 1) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘((𝑁 − 1) + 1)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘((𝑁 − 1) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘((𝑁 − 1) + 1)))) |
4 | eluzelcn 12858 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ℂ) | |
5 | ax-1cn 11190 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 11493 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 585 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 481 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | fveq2d 6895 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘((𝑁 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑁)) |
10 | 8 | fveq2d 6895 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘((𝑁 − 1) + 1)) = (𝐹‘𝑁)) |
11 | 10 | oveq2d 7430 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘((𝑁 − 1) + 1))) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
12 | 3, 9, 11 | 3eqtr3d 2775 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 1c1 11133 + caddc 11135 − cmin 11468 ℤcz 12582 ℤ≥cuz 12846 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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-seq 13993 |
This theorem is referenced by: seqf1olem2 14033 seqid 14038 seqz 14041 bcn2 14304 seqcoll 14451 serf0 15653 lgsval2lem 27233 cvmliftlem5 34889 |
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