Theorem prdsmndd 18964

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 2762	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2762	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
3		prdsmndd.y	. . . 4 ⊢ 𝑌 = (𝑆Xs𝑅)
4		eqid 2761	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
5		eqid 2761	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
6		prdsmndd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7	6	elexd 3474	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9		prdsmndd.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
10	9	elexd 3474	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12		prdsmndd.r	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
13	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
14		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
15		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
16	3, 4, 5, 8, 11, 13, 14, 15	prdsplusgcl 18962	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
17	16	3impb 1132	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
18	12	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd)
19	18	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd)
20	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ V)
21	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
22	12	ffnd 6710	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
23	22	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
24		simplr1 1234	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
25		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
26	3, 4, 20, 21, 23, 24, 25	prdsbasprj 17643	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
27		simplr2 1235	. . . . . . 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 17643	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
29		simplr3 1236	. . . . . . 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 17643	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
31		eqid 2761	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
32		eqid 2761	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
33	31, 32	mndass 18932	. . . . . 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 1399	. . . . 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 17645	. . . . . 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 7435	. . . . 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 17645	. . . . . 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 7436	. . . . 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 2806	. . . 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 5198	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
42	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
43	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44	16	3adantr3 1190	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌))
45		simpr3 1215	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
46	3, 4, 41, 42, 43, 44, 45, 5	prdsplusgval 17644	. . 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 1213	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
48	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
49		simpr2 1214	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
50	3, 4, 5, 41, 42, 48, 49, 45	prdsplusgcl 18962	. . . 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 17644	. . 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 2806	. 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 2761	. . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅)
54	3, 4, 5, 7, 10, 12, 53	prdsidlem 18963	. . 3 ⊢ (𝜑 → ((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)))
55	54	simpld 500	. 2 ⊢ (𝜑 → (0g ∘ 𝑅) ∈ (Base‘𝑌))
56	54	simprd 501	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎))
57	56	r19.21bi 3255	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎))
58	57	simpld 500	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎)
59	57	simprd 501	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)
60	1, 2, 17, 52, 55, 58, 59	ismndd 18946	1 ⊢ (𝜑 → 𝑌 ∈ Mnd)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ↦ cmpt 5186   ∘ ccom 5655   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  X_scprds 17616  Mndcmnd 18923

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-0g 17612  df-prds 17618  df-mgm 18816  df-sgrp 18908  df-mnd 18924

This theorem is used by:  prds0g  18965  pwsmnd  18966  xpsmnd  18971  prdspjmhm  19025  prdsgrpd  19260  prdscmnd  20075  prdsringd  20550  dsmm0cl  22046  prdstmdd  24443