Theorem prodtp 29573

Description: A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)

Hypotheses

Ref	Expression
prodpr.1	|- ( k = A -> D = E )
prodpr.2	|- ( k = B -> D = F )
prodpr.a	|- ( ph -> A e. V )
prodpr.b	|- ( ph -> B e. W )
prodpr.e	|- ( ph -> E e. CC )
prodpr.f	|- ( ph -> F e. CC )
prodpr.3	|- ( ph -> A =/= B )
prodtp.1	|- ( k = C -> D = G )
prodtp.c	|- ( ph -> C e. X )
prodtp.g	|- ( ph -> G e. CC )
prodtp.2	|- ( ph -> A =/= C )
prodtp.3	|- ( ph -> B =/= C )

Assertion

Ref	Expression
prodtp	|- ( ph -> prod_ k e. { A , B , C } D = ( ( E x. F ) x. G ) )

Distinct variable groups:   A, k   B, k   C, k   k, E   k, F   k, G   k, V   k, W   k, X   ph, k

Allowed substitution hint:   D( k)

Proof of Theorem prodtp

Step	Hyp	Ref	Expression
1		prodtp.2	. . . 4 |- ( ph -> A =/= C )
2		prodtp.3	. . . 4 |- ( ph -> B =/= C )
3		disjprsn 4250	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> ( { A , B } i^i { C } ) = (/) )
5		df-tp 4182	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
6	5	a1i 11	. . 3 |- ( ph -> { A , B , C } = ( { A , B } u. { C } ) )
7		tpfi 8236	. . . 4 |- { A , B , C } e. Fin
8	7	a1i 11	. . 3 |- ( ph -> { A , B , C } e. Fin )
9		vex 3203	. . . . 5 |- k e. _V
10	9	eltp 4230	. . . 4 |- ( k e. { A , B , C } <-> ( k = A \/ k = B \/ k = C ) )
11		prodpr.1	. . . . . . . 8 |- ( k = A -> D = E )
12	11	adantl 482	. . . . . . 7 |- ( ( ph /\ k = A ) -> D = E )
13		prodpr.e	. . . . . . . 8 |- ( ph -> E e. CC )
14	13	adantr 481	. . . . . . 7 |- ( ( ph /\ k = A ) -> E e. CC )
15	12, 14	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ k = A ) -> D e. CC )
16	15	adantlr 751	. . . . 5 |- ( ( ( ph /\ ( k = A \/ k = B \/ k = C ) ) /\ k = A ) -> D e. CC )
17		prodpr.2	. . . . . . . 8 |- ( k = B -> D = F )
18	17	adantl 482	. . . . . . 7 |- ( ( ph /\ k = B ) -> D = F )
19		prodpr.f	. . . . . . . 8 |- ( ph -> F e. CC )
20	19	adantr 481	. . . . . . 7 |- ( ( ph /\ k = B ) -> F e. CC )
21	18, 20	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ k = B ) -> D e. CC )
22	21	adantlr 751	. . . . 5 |- ( ( ( ph /\ ( k = A \/ k = B \/ k = C ) ) /\ k = B ) -> D e. CC )
23		prodtp.1	. . . . . . . 8 |- ( k = C -> D = G )
24	23	adantl 482	. . . . . . 7 |- ( ( ph /\ k = C ) -> D = G )
25		prodtp.g	. . . . . . . 8 |- ( ph -> G e. CC )
26	25	adantr 481	. . . . . . 7 |- ( ( ph /\ k = C ) -> G e. CC )
27	24, 26	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ k = C ) -> D e. CC )
28	27	adantlr 751	. . . . 5 |- ( ( ( ph /\ ( k = A \/ k = B \/ k = C ) ) /\ k = C ) -> D e. CC )
29		simpr 477	. . . . 5 |- ( ( ph /\ ( k = A \/ k = B \/ k = C ) ) -> ( k = A \/ k = B \/ k = C ) )
30	16, 22, 28, 29	mpjao3dan 1395	. . . 4 |- ( ( ph /\ ( k = A \/ k = B \/ k = C ) ) -> D e. CC )
31	10, 30	sylan2b 492	. . 3 |- ( ( ph /\ k e. { A , B , C } ) -> D e. CC )
32	4, 6, 8, 31	fprodsplit 14696	. 2 |- ( ph -> prod_ k e. { A , B , C } D = ( prod_ k e. { A , B } D x. prod_ k e. { C } D ) )
33		prodpr.a	. . . 4 |- ( ph -> A e. V )
34		prodpr.b	. . . 4 |- ( ph -> B e. W )
35		prodpr.3	. . . 4 |- ( ph -> A =/= B )
36	11, 17, 33, 34, 13, 19, 35	prodpr 29572	. . 3 |- ( ph -> prod_ k e. { A , B } D = ( E x. F ) )
37		prodtp.c	. . . 4 |- ( ph -> C e. X )
38	23	prodsn 14692	. . . 4 |- ( ( C e. X /\ G e. CC ) -> prod_ k e. { C } D = G )
39	37, 25, 38	syl2anc 693	. . 3 |- ( ph -> prod_ k e. { C } D = G )
40	36, 39	oveq12d 6668	. 2 |- ( ph -> ( prod_ k e. { A , B } D x. prod_ k e. { C } D ) = ( ( E x. F ) x. G ) )
41	32, 40	eqtrd 2656	1 |- ( ph -> prod_ k e. { A , B , C } D = ( ( E x. F ) x. G ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181  (class class class)co 6650   Fincfn 7955   CCcc 9934   x. cmul 9941   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636

This theorem is referenced by:  hgt750lemg  30732