Theorem prdssgrpd 18692

Description: The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)

Hypotheses

Ref	Expression
prdssgrpd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdssgrpd.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdssgrpd.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdssgrpd.r	⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)

Assertion

Ref	Expression
prdssgrpd	⊢ (𝜑 → 𝑌 ∈ Smgrp)

Proof of Theorem prdssgrpd

Dummy variables 𝑦 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2726	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2726	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
3		prdssgrpd.y	. . . 4 ⊢ 𝑌 = (𝑆Xs𝑅)
4		eqid 2725	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
5		eqid 2725	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
6		prdssgrpd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7	6	elexd 3484	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
8	7	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9		prdssgrpd.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
10	9	elexd 3484	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
11	10	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12		prdssgrpd.r	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)
13	12	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
14		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
15		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
16	3, 4, 5, 8, 11, 13, 14, 15	prdsplusgsgrpcl 18691	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
17	16	3impb 1112	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
18	12	ffvelcdmda 7089	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Smgrp)
19	18	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Smgrp)
20	7	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ V)
21	10	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
22	12	ffnd 6718	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
23	22	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
24		simplr1 1212	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
25		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
26	3, 4, 20, 21, 23, 24, 25	prdsbasprj 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
27		simplr2 1213	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌))
28	3, 4, 20, 21, 23, 27, 25	prdsbasprj 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
29		simplr3 1214	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌))
30	3, 4, 20, 21, 23, 29, 25	prdsbasprj 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
31		eqid 2725	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
32		eqid 2725	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
33	31, 32	sgrpass 18684	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ Smgrp ∧ ((𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))))
34	19, 26, 28, 30, 33	syl13anc 1369	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))))
35	3, 4, 20, 21, 23, 24, 27, 5, 25	prdsplusgfval 17455	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘𝑌)𝑏)‘𝑦) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦)))
36	35	oveq1d 7431	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))
37	3, 4, 20, 21, 23, 27, 29, 5, 25	prdsplusgfval 17455	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏(+g‘𝑌)𝑐)‘𝑦) = ((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))
38	37	oveq2d 7432	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))))
39	34, 36, 38	3eqtr4d 2775	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)))
40	39	mpteq2dva 5243	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))))
41	7	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
42	10	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
43	22	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44	16	3adantr3 1168	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
45		simpr3 1193	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
46	3, 4, 41, 42, 43, 44, 45, 5	prdsplusgval 17454	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑌)𝑏)(+g‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))))
47		simpr1 1191	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
48	12	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
49		simpr2 1192	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
50	3, 4, 5, 41, 42, 48, 49, 45	prdsplusgsgrpcl 18691	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌))
51	3, 4, 41, 42, 43, 47, 50, 5	prdsplusgval 17454	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)(𝑏(+g‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))))
52	40, 46, 51	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑌)𝑏)(+g‘𝑌)𝑐) = (𝑎(+g‘𝑌)(𝑏(+g‘𝑌)𝑐)))
53	3	ovexi 7450	. . 3 ⊢ 𝑌 ∈ V
54	53	a1i 11	. 2 ⊢ (𝜑 → 𝑌 ∈ V)
55	1, 2, 17, 52, 54	issgrpd 18689	1 ⊢ (𝜑 → 𝑌 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ↦ cmpt 5226   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  +_gcplusg 17232  X_scprds 17426  Smgrpcsgrp 18677

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-hom 17256  df-cco 17257  df-prds 17428  df-mgm 18599  df-sgrp 18678

This theorem is referenced by:  prdsrngd  20120