Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsmndd Structured version   Visualization version   GIF version

Theorem prdsmndd 17943
 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 (𝜑𝐼𝑊)
prdsmndd.s (𝜑𝑆𝑉)
prdsmndd.r (𝜑𝑅:𝐼⟶Mnd)
Assertion
Ref Expression
prdsmndd (𝜑𝑌 ∈ Mnd)

Proof of Theorem prdsmndd
Dummy variables 𝑎 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2799 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2799 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdsmndd.y . . . 4 𝑌 = (𝑆Xs𝑅)
4 eqid 2798 . . . 4 (Base‘𝑌) = (Base‘𝑌)
5 eqid 2798 . . . 4 (+g𝑌) = (+g𝑌)
6 prdsmndd.s . . . . . 6 (𝜑𝑆𝑉)
76elexd 3461 . . . . 5 (𝜑𝑆 ∈ V)
87adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9 prdsmndd.i . . . . . 6 (𝜑𝐼𝑊)
109elexd 3461 . . . . 5 (𝜑𝐼 ∈ V)
1110adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12 prdsmndd.r . . . . 5 (𝜑𝑅:𝐼⟶Mnd)
1312adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
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))
163, 4, 5, 8, 11, 13, 14, 15prdsplusgcl 17941 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
17163impb 1112 . 2 ((𝜑𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
1812ffvelrnda 6833 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
1918adantlr 714 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
207ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ V)
2110ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
2212ffnd 6491 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
2322ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
24 simplr1 1212 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
25 simpr 488 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
263, 4, 20, 21, 23, 24, 25prdsbasprj 16744 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
27 simplr2 1213 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
283, 4, 20, 21, 23, 27, 25prdsbasprj 16744 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
29 simplr3 1214 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
303, 4, 20, 21, 23, 29, 25prdsbasprj 16744 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
31 eqid 2798 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
32 eqid 2798 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
3331, 32mndass 17919 . . . . . 6 (((𝑅𝑦) ∈ Mnd ∧ ((𝑎𝑦) ∈ (Base‘(𝑅𝑦)) ∧ (𝑏𝑦) ∈ (Base‘(𝑅𝑦)) ∧ (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))) → (((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
3419, 26, 28, 30, 33syl13anc 1369 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
353, 4, 20, 21, 23, 24, 27, 5, 25prdsplusgfval 16746 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑌)𝑏)‘𝑦) = ((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦)))
3635oveq1d 7155 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (((𝑎(+g𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)) = (((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑐𝑦)))
373, 4, 20, 21, 23, 27, 29, 5, 25prdsplusgfval 16746 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
3837oveq2d 7156 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦)) = ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
3934, 36, 383eqtr4d 2843 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (((𝑎(+g𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦)))
4039mpteq2dva 5126 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦𝐼 ↦ (((𝑎(+g𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))) = (𝑦𝐼 ↦ ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
417adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
4210adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
4322adantr 484 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44163adantr3 1168 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
45 simpr3 1193 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
463, 4, 41, 42, 43, 44, 45, 5prdsplusgval 16745 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑦𝐼 ↦ (((𝑎(+g𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
47 simpr1 1191 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
4812adantr 484 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
49 simpr2 1192 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
503, 4, 5, 41, 42, 48, 49, 45prdsplusgcl 17941 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
513, 4, 41, 42, 43, 47, 50, 5prdsplusgval 16745 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)) = (𝑦𝐼 ↦ ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
5240, 46, 513eqtr4d 2843 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)))
53 eqid 2798 . . . 4 (0g𝑅) = (0g𝑅)
543, 4, 5, 7, 10, 12, 53prdsidlem 17942 . . 3 (𝜑 → ((0g𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎)))
5554simpld 498 . 2 (𝜑 → (0g𝑅) ∈ (Base‘𝑌))
5654simprd 499 . . . 4 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
5756r19.21bi 3173 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
5857simpld 498 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((0g𝑅)(+g𝑌)𝑎) = 𝑎)
5957simprd 499 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)(0g𝑅)) = 𝑎)
601, 2, 17, 52, 55, 58, 59ismndd 17932 1 (𝜑𝑌 ∈ Mnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ↦ cmpt 5111   ∘ ccom 5524   Fn wfn 6322  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140  Basecbs 16482  +gcplusg 16564  0gc0g 16712  Xscprds 16718  Mndcmnd 17910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-map 8398  df-ixp 8452  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-sup 8897  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-nn 11633  df-2 11695  df-3 11696  df-4 11697  df-5 11698  df-6 11699  df-7 11700  df-8 11701  df-9 11702  df-n0 11893  df-z 11977  df-dec 12094  df-uz 12239  df-fz 12893  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-plusg 16577  df-mulr 16578  df-sca 16580  df-vsca 16581  df-ip 16582  df-tset 16583  df-ple 16584  df-ds 16586  df-hom 16588  df-cco 16589  df-0g 16714  df-prds 16720  df-mgm 17851  df-sgrp 17900  df-mnd 17911 This theorem is referenced by:  prds0g  17944  pwsmnd  17945  xpsmnd  17950  prdspjmhm  17992  prdsgrpd  18209  prdscmnd  18982  prdsringd  19366  dsmm0cl  20438  prdstmdd  22743
 Copyright terms: Public domain W3C validator