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Mirrors > Home > ILE Home > Th. List > prod0 | GIF version |
Description: A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prod0 | ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9176 | . 2 ⊢ 1 ∈ ℤ | |
2 | nnuz 9457 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | id 19 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℤ) | |
4 | 1ap0 8448 | . . . 4 ⊢ 1 # 0 | |
5 | 4 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → 1 # 0) |
6 | 2 | prodfclim1 11423 | . . 3 ⊢ (1 ∈ ℤ → seq1( · , (ℕ × {1})) ⇝ 1) |
7 | noel 3398 | . . . . . . 7 ⊢ ¬ 𝑗 ∈ ∅ | |
8 | 7 | olci 722 | . . . . . 6 ⊢ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅) |
9 | df-dc 821 | . . . . . 6 ⊢ (DECID 𝑗 ∈ ∅ ↔ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅)) | |
10 | 8, 9 | mpbir 145 | . . . . 5 ⊢ DECID 𝑗 ∈ ∅ |
11 | 10 | rgenw 2512 | . . . 4 ⊢ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅ |
12 | 11 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅) |
13 | 0ss 3432 | . . . 4 ⊢ ∅ ⊆ ℕ | |
14 | 13 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∅ ⊆ ℕ) |
15 | fvconst2g 5678 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = 1) | |
16 | noel 3398 | . . . . 5 ⊢ ¬ 𝑘 ∈ ∅ | |
17 | 16 | iffalsei 3514 | . . . 4 ⊢ if(𝑘 ∈ ∅, 𝐴, 1) = 1 |
18 | 15, 17 | eqtr4di 2208 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 1)) |
19 | 16 | pm2.21i 636 | . . . 4 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
20 | 19 | adantl 275 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
21 | 2, 3, 5, 6, 12, 14, 18, 20 | zprodap0 11460 | . 2 ⊢ (1 ∈ ℤ → ∏𝑘 ∈ ∅ 𝐴 = 1) |
22 | 1, 21 | ax-mp 5 | 1 ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 698 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 ∅c0 3394 ifcif 3505 {csn 3560 class class class wbr 3965 × cxp 4581 ‘cfv 5167 ℂcc 7713 0cc0 7715 1c1 7716 # cap 8439 ℕcn 8816 ℤcz 9150 ∏cprod 11429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-iinf 4545 ax-cnex 7806 ax-resscn 7807 ax-1cn 7808 ax-1re 7809 ax-icn 7810 ax-addcl 7811 ax-addrcl 7812 ax-mulcl 7813 ax-mulrcl 7814 ax-addcom 7815 ax-mulcom 7816 ax-addass 7817 ax-mulass 7818 ax-distr 7819 ax-i2m1 7820 ax-0lt1 7821 ax-1rid 7822 ax-0id 7823 ax-rnegex 7824 ax-precex 7825 ax-cnre 7826 ax-pre-ltirr 7827 ax-pre-ltwlin 7828 ax-pre-lttrn 7829 ax-pre-apti 7830 ax-pre-ltadd 7831 ax-pre-mulgt0 7832 ax-pre-mulext 7833 ax-arch 7834 ax-caucvg 7835 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4252 df-po 4255 df-iso 4256 df-iord 4325 df-on 4327 df-ilim 4328 df-suc 4330 df-iom 4548 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-res 4595 df-ima 4596 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-f1 5172 df-fo 5173 df-f1o 5174 df-fv 5175 df-isom 5176 df-riota 5774 df-ov 5821 df-oprab 5822 df-mpo 5823 df-1st 6082 df-2nd 6083 df-recs 6246 df-irdg 6311 df-frec 6332 df-1o 6357 df-oadd 6361 df-er 6473 df-en 6679 df-dom 6680 df-fin 6681 df-pnf 7897 df-mnf 7898 df-xr 7899 df-ltxr 7900 df-le 7901 df-sub 8031 df-neg 8032 df-reap 8433 df-ap 8440 df-div 8529 df-inn 8817 df-2 8875 df-3 8876 df-4 8877 df-n0 9074 df-z 9151 df-uz 9423 df-q 9511 df-rp 9543 df-fz 9895 df-fzo 10024 df-seqfrec 10327 df-exp 10401 df-ihash 10632 df-cj 10724 df-re 10725 df-im 10726 df-rsqrt 10880 df-abs 10881 df-clim 11158 df-proddc 11430 |
This theorem is referenced by: prod1dc 11465 fprodf1o 11467 fprodmul 11470 fprodcl2lem 11484 fprodcllem 11485 fprodfac 11494 fprodconst 11499 fprodap0 11500 fprod2d 11502 fprodrec 11508 fprodap0f 11515 fprodle 11519 fprodmodd 11520 |
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