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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 9238 | . 2 ⊢ 1 ∈ ℤ | |
2 | nnuz 9522 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | id 19 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℤ) | |
4 | 1ap0 8509 | . . . 4 ⊢ 1 # 0 | |
5 | 4 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → 1 # 0) |
6 | 2 | prodfclim1 11507 | . . 3 ⊢ (1 ∈ ℤ → seq1( · , (ℕ × {1})) ⇝ 1) |
7 | noel 3418 | . . . . . . 7 ⊢ ¬ 𝑗 ∈ ∅ | |
8 | 7 | olci 727 | . . . . . 6 ⊢ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅) |
9 | df-dc 830 | . . . . . 6 ⊢ (DECID 𝑗 ∈ ∅ ↔ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅)) | |
10 | 8, 9 | mpbir 145 | . . . . 5 ⊢ DECID 𝑗 ∈ ∅ |
11 | 10 | rgenw 2525 | . . . 4 ⊢ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅ |
12 | 11 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅) |
13 | 0ss 3453 | . . . 4 ⊢ ∅ ⊆ ℕ | |
14 | 13 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∅ ⊆ ℕ) |
15 | fvconst2g 5710 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = 1) | |
16 | noel 3418 | . . . . 5 ⊢ ¬ 𝑘 ∈ ∅ | |
17 | 16 | iffalsei 3535 | . . . 4 ⊢ if(𝑘 ∈ ∅, 𝐴, 1) = 1 |
18 | 15, 17 | eqtr4di 2221 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 1)) |
19 | 16 | pm2.21i 641 | . . . 4 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
20 | 19 | adantl 275 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
21 | 2, 3, 5, 6, 12, 14, 18, 20 | zprodap0 11544 | . 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 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ∅c0 3414 ifcif 3526 {csn 3583 class class class wbr 3989 × cxp 4609 ‘cfv 5198 ℂcc 7772 0cc0 7774 1c1 7775 # cap 8500 ℕcn 8878 ℤcz 9212 ∏cprod 11513 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 |
This theorem is referenced by: prod1dc 11549 fprodf1o 11551 fprodmul 11554 fprodcl2lem 11568 fprodcllem 11569 fprodfac 11578 fprodconst 11583 fprodap0 11584 fprod2d 11586 fprodrec 11592 fprodap0f 11599 fprodle 11603 fprodmodd 11604 |
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