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Mirrors > Home > ILE Home > Th. List > seq3m1 | GIF version |
Description: Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.) |
Ref | Expression |
---|---|
seq3m1.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seq3m1.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
seq3m1.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seq3m1.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seq3m1 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3m1.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | seq3m1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) | |
3 | eluzp1m1 9096 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) | |
4 | 1, 2, 3 | syl2anc 404 | . . 3 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) |
5 | seq3m1.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
6 | seq3m1.pl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
7 | 4, 5, 6 | seq3p1 9938 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘((𝑁 − 1) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘((𝑁 − 1) + 1)))) |
8 | eluzelcn 9084 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ℂ) | |
9 | ax-1cn 7492 | . . . . 5 ⊢ 1 ∈ ℂ | |
10 | npcan 7745 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
11 | 8, 9, 10 | sylancl 405 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → ((𝑁 − 1) + 1) = 𝑁) |
12 | 2, 11 | syl 14 | . . 3 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
13 | 12 | fveq2d 5322 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘((𝑁 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑁)) |
14 | 12 | fveq2d 5322 | . . 3 ⊢ (𝜑 → (𝐹‘((𝑁 − 1) + 1)) = (𝐹‘𝑁)) |
15 | 14 | oveq2d 5682 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘((𝑁 − 1) + 1))) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
16 | 7, 13, 15 | 3eqtr3d 2129 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 ‘cfv 5028 (class class class)co 5666 ℂcc 7402 1c1 7405 + caddc 7407 − cmin 7707 ℤcz 8804 ℤ≥cuz 9073 seqcseq 9906 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-inn 8477 df-n0 8728 df-z 8805 df-uz 9074 df-iseq 9907 df-seq3 9908 |
This theorem is referenced by: seq3f1olemqsumkj 9981 serf0 10795 |
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