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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 9277 | . 2 ⊢ 1 ∈ ℤ | |
2 | nnuz 9561 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | id 19 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℤ) | |
4 | 1ap0 8545 | . . . 4 ⊢ 1 # 0 | |
5 | 4 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → 1 # 0) |
6 | 2 | prodfclim1 11547 | . . 3 ⊢ (1 ∈ ℤ → seq1( · , (ℕ × {1})) ⇝ 1) |
7 | noel 3426 | . . . . . . 7 ⊢ ¬ 𝑗 ∈ ∅ | |
8 | 7 | olci 732 | . . . . . 6 ⊢ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅) |
9 | df-dc 835 | . . . . . 6 ⊢ (DECID 𝑗 ∈ ∅ ↔ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅)) | |
10 | 8, 9 | mpbir 146 | . . . . 5 ⊢ DECID 𝑗 ∈ ∅ |
11 | 10 | rgenw 2532 | . . . 4 ⊢ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅ |
12 | 11 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅) |
13 | 0ss 3461 | . . . 4 ⊢ ∅ ⊆ ℕ | |
14 | 13 | a1i 9 | . . 3 ⊢ (1 ∈ ℤ → ∅ ⊆ ℕ) |
15 | fvconst2g 5730 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = 1) | |
16 | noel 3426 | . . . . 5 ⊢ ¬ 𝑘 ∈ ∅ | |
17 | 16 | iffalsei 3543 | . . . 4 ⊢ if(𝑘 ∈ ∅, 𝐴, 1) = 1 |
18 | 15, 17 | eqtr4di 2228 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 1)) |
19 | 16 | pm2.21i 646 | . . . 4 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
20 | 19 | adantl 277 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
21 | 2, 3, 5, 6, 12, 14, 18, 20 | zprodap0 11584 | . 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 104 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 ifcif 3534 {csn 3592 class class class wbr 4003 × cxp 4624 ‘cfv 5216 ℂcc 7808 0cc0 7810 1c1 7811 # cap 8536 ℕcn 8917 ℤcz 9251 ∏cprod 11553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-ihash 10751 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-proddc 11554 |
This theorem is referenced by: prod1dc 11589 fprodf1o 11591 fprodmul 11594 fprodcl2lem 11608 fprodcllem 11609 fprodfac 11618 fprodconst 11623 fprodap0 11624 fprod2d 11626 fprodrec 11632 fprodap0f 11639 fprodle 11643 fprodmodd 11644 |
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