fallfac1 - Metamath Proof Explorer

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Theorem fallfac1 15978

Description: The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)

**Assertion**
Ref	Expression
fallfac1	⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)

**Proof of Theorem fallfac1**
Step	Hyp	Ref	Expression
1		0p1e1 12334	. . 3 ⊢ (0 + 1) = 1
2	1	oveq2i 7420	. 2 ⊢ (𝐴 FallFac (0 + 1)) = (𝐴 FallFac 1)
3		0nn0 12487	. . . 4 ⊢ 0 ∈ ℕ₀
4		fallfacp1 15974	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝐴 FallFac (0 + 1)) = ((𝐴 FallFac 0) · (𝐴 − 0)))
5	3, 4	mpan2 690	. . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac (0 + 1)) = ((𝐴 FallFac 0) · (𝐴 − 0)))
6		fallfac0 15972	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
7		subid1 11480	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
8	6, 7	oveq12d 7427	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 FallFac 0) · (𝐴 − 0)) = (1 · 𝐴))
9		mullid 11213	. . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
10	5, 8, 9	3eqtrd 2777	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac (0 + 1)) = 𝐴)
11	2, 10	eqtr3id 2787	1 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 ℕ₀cn0 12472 FallFac cfallfac 15948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-prod 15850 df-risefac 15950 df-fallfac 15951

This theorem is referenced by: (None)

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