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Mirrors > Home > ILE Home > Th. List > iseqid3 | GIF version |
Description: A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.) |
Ref | Expression |
---|---|
iseqid3.1 | ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) |
iseqid3.2 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
iseqid3.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = 𝑍) |
iseqid3.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
iseqid3 | ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqid3.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | iseqid3.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = 𝑍) | |
3 | iseqid3.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
4 | elsn2g 3435 | . . . . . 6 ⊢ (𝑍 ∈ 𝑉 → ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)) |
6 | 5 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)) |
7 | 2, 6 | mpbird 165 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ {𝑍}) |
8 | iseqid3.1 | . . . . . 6 ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) | |
9 | elsn2g 3435 | . . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → ((𝑍 + 𝑍) ∈ {𝑍} ↔ (𝑍 + 𝑍) = 𝑍)) | |
10 | 3, 9 | syl 14 | . . . . . 6 ⊢ (𝜑 → ((𝑍 + 𝑍) ∈ {𝑍} ↔ (𝑍 + 𝑍) = 𝑍)) |
11 | 8, 10 | mpbird 165 | . . . . 5 ⊢ (𝜑 → (𝑍 + 𝑍) ∈ {𝑍}) |
12 | elsni 3424 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑍} → 𝑥 = 𝑍) | |
13 | elsni 3424 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑍} → 𝑦 = 𝑍) | |
14 | 12, 13 | oveqan12d 5562 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → (𝑥 + 𝑦) = (𝑍 + 𝑍)) |
15 | 14 | eleq1d 2148 | . . . . 5 ⊢ ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → ((𝑥 + 𝑦) ∈ {𝑍} ↔ (𝑍 + 𝑍) ∈ {𝑍})) |
16 | 11, 15 | syl5ibrcom 155 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → (𝑥 + 𝑦) ∈ {𝑍})) |
17 | 16 | imp 122 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍})) → (𝑥 + 𝑦) ∈ {𝑍}) |
18 | 1, 7, 17 | iseqcl 9537 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) ∈ {𝑍}) |
19 | elsni 3424 | . 2 ⊢ ((seq𝑀( + , 𝐹, {𝑍})‘𝑁) ∈ {𝑍} → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍) | |
20 | 18, 19 | syl 14 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 {csn 3406 ‘cfv 4932 (class class class)co 5543 ℤ≥cuz 8700 seqcseq 9521 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-ltadd 7154 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-ilim 4132 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-frec 6040 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-inn 8107 df-n0 8356 df-z 8433 df-uz 8701 df-iseq 9522 |
This theorem is referenced by: (None) |
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