Theorem prdsmndd 17944

Description: The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
prdsmndd	⊢ (𝜑 → 𝑌 ∈ Mnd)

Proof of Theorem prdsmndd

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

Step	Hyp	Ref	Expression
1		eqidd 2822	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2822	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
3		prdsmndd.y	. . . 4 ⊢ 𝑌 = (𝑆Xs𝑅)
4		eqid 2821	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
5		eqid 2821	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
6		prdsmndd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7	6	elexd 3514	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
8	7	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9		prdsmndd.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
10	9	elexd 3514	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
11	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12		prdsmndd.r	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
13	12	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
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	prdsplusgcl 17942	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
17	16	3impb 1111	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
18	12	ffvelrnda 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd)
19	18	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd)
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 6515	. . . . . . . 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 1211	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
25		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
26	3, 4, 20, 21, 23, 24, 25	prdsbasprj 16745	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
27		simplr2 1212	. . . . . . 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 16745	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
29		simplr3 1213	. . . . . . 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 16745	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
31		eqid 2821	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
32		eqid 2821	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
33	31, 32	mndass 17920	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ Mnd ∧ ((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 1368	. . . . 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 16747	. . . . . 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 7171	. . . . 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 16747	. . . . . 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 7172	. . . . 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 2866	. . . 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 5161	. . 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 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
42	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
43	22	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44	16	3adantr3 1167	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
45		simpr3 1192	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
46	3, 4, 41, 42, 43, 44, 45, 5	prdsplusgval 16746	. . 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 1190	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
48	12	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
49		simpr2 1191	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
50	3, 4, 5, 41, 42, 48, 49, 45	prdsplusgcl 17942	. . . 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 16746	. . 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 2866	. 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		eqid 2821	. . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅)
54	3, 4, 5, 7, 10, 12, 53	prdsidlem 17943	. . 3 ⊢ (𝜑 → ((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)))
55	54	simpld 497	. 2 ⊢ (𝜑 → (0g ∘ 𝑅) ∈ (Base‘𝑌))
56	54	simprd 498	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎))
57	56	r19.21bi 3208	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎))
58	57	simpld 497	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎)
59	57	simprd 498	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)
60	1, 2, 17, 52, 55, 58, 59	ismndd 17933	1 ⊢ (𝜑 → 𝑌 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ↦ cmpt 5146   ∘ ccom 5559   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  X_scprds 16719  Mndcmnd 17911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-0g 16715  df-prds 16721  df-mgm 17852  df-sgrp 17901  df-mnd 17912

This theorem is referenced by:  prds0g  17945  pwsmnd  17946  xpsmnd  17951  prdspjmhm  17993  prdsgrpd  18209  prdscmnd  18981  prdsringd  19362  dsmm0cl  20884  prdstmdd  22732