Theorem prdssgrpd 18660

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 2730	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2730	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
3		prdssgrpd.y	. . . 4 ⊢ 𝑌 = (𝑆Xs𝑅)
4		eqid 2729	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
5		eqid 2729	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
6		prdssgrpd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7	6	elexd 3471	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9		prdssgrpd.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
10	9	elexd 3471	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12		prdssgrpd.r	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
14		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
15		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
16	3, 4, 5, 8, 11, 13, 14, 15	prdsplusgsgrpcl 18659	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
17	16	3impb 1114	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
18	12	ffvelcdmda 7056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Smgrp)
19	18	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Smgrp)
20	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ V)
21	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
22	12	ffnd 6689	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
23	22	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
24		simplr1 1216	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
25		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
26	3, 4, 20, 21, 23, 24, 25	prdsbasprj 17435	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
27		simplr2 1217	. . . . . . 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 17435	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
29		simplr3 1218	. . . . . . 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 17435	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
31		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
32		eqid 2729	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
33	31, 32	sgrpass 18652	. . . . . 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 1374	. . . . 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 17437	. . . . . 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 7402	. . . . 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 17437	. . . . . 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 7403	. . . . 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 2774	. . . 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 5200	. . 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 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
42	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
43	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44	16	3adantr3 1172	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
45		simpr3 1197	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
46	3, 4, 41, 42, 43, 44, 45, 5	prdsplusgval 17436	. . 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 1195	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
48	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
49		simpr2 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
50	3, 4, 5, 41, 42, 48, 49, 45	prdsplusgsgrpcl 18659	. . . 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 17436	. . 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 2774	. 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 7421	. . 3 ⊢ 𝑌 ∈ V
54	53	a1i 11	. 2 ⊢ (𝜑 → 𝑌 ∈ V)
55	1, 2, 17, 52, 54	issgrpd 18657	1 ⊢ (𝜑 → 𝑌 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ↦ cmpt 5188   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  X_scprds 17408  Smgrpcsgrp 18645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-prds 17410  df-mgm 18567  df-sgrp 18646

This theorem is referenced by:  prdsrngd  20085