fallfac1 - Metamath Proof Explorer

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Theorem fallfac1 15981

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 12335	. . 3 ⊢ (0 + 1) = 1
2	1	oveq2i 7415	. 2 ⊢ (𝐴 FallFac (0 + 1)) = (𝐴 FallFac 1)
3		0nn0 12488	. . . 4 ⊢ 0 ∈ ℕ₀
4		fallfacp1 15977	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝐴 FallFac (0 + 1)) = ((𝐴 FallFac 0) · (𝐴 − 0)))
5	3, 4	mpan2 688	. . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac (0 + 1)) = ((𝐴 FallFac 0) · (𝐴 − 0)))
6		fallfac0 15975	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
7		subid1 11481	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
8	6, 7	oveq12d 7422	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 FallFac 0) · (𝐴 − 0)) = (1 · 𝐴))
9		mullid 11214	. . 3 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
10	5, 8, 9	3eqtrd 2770	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac (0 + 1)) = 𝐴)
11	2, 10	eqtr3id 2780	1 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7404 ℂcc 11107 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 − cmin 11445 ℕ₀cn0 12473 FallFac cfallfac 15951

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-prod 15853 df-risefac 15953 df-fallfac 15954

This theorem is referenced by: (None)

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