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Theorem prdsmndd 18635
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 2732 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2732 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdsmndd.y . . . 4 𝑌 = (𝑆Xs𝑅)
4 eqid 2731 . . . 4 (Base‘𝑌) = (Base‘𝑌)
5 eqid 2731 . . . 4 (+g𝑌) = (+g𝑌)
6 prdsmndd.s . . . . . 6 (𝜑𝑆𝑉)
76elexd 3493 . . . . 5 (𝜑𝑆 ∈ V)
87adantr 481 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9 prdsmndd.i . . . . . 6 (𝜑𝐼𝑊)
109elexd 3493 . . . . 5 (𝜑𝐼 ∈ V)
1110adantr 481 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12 prdsmndd.r . . . . 5 (𝜑𝑅:𝐼⟶Mnd)
1312adantr 481 . . . 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))
163, 4, 5, 8, 11, 13, 14, 15prdsplusgcl 18633 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
17163impb 1115 . 2 ((𝜑𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
1812ffvelcdmda 7071 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
1918adantlr 713 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
207ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ V)
2110ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
2212ffnd 6705 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
2322ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
24 simplr1 1215 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
25 simpr 485 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
263, 4, 20, 21, 23, 24, 25prdsbasprj 17400 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
27 simplr2 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
283, 4, 20, 21, 23, 27, 25prdsbasprj 17400 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
29 simplr3 1217 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
303, 4, 20, 21, 23, 29, 25prdsbasprj 17400 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
31 eqid 2731 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
32 eqid 2731 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
3331, 32mndass 18611 . . . . . 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 1372 . . . . 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 17402 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑌)𝑏)‘𝑦) = ((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦)))
3635oveq1d 7408 . . . . 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 17402 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
3837oveq2d 7409 . . . . 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 2781 . . . 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 5241 . . 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 481 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
4210adantr 481 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
4322adantr 481 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44163adantr3 1171 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
45 simpr3 1196 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
463, 4, 41, 42, 43, 44, 45, 5prdsplusgval 17401 . . 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 1194 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
4812adantr 481 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
49 simpr2 1195 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
503, 4, 5, 41, 42, 48, 49, 45prdsplusgcl 18633 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
513, 4, 41, 42, 43, 47, 50, 5prdsplusgval 17401 . . 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 2781 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)))
53 eqid 2731 . . . 4 (0g𝑅) = (0g𝑅)
543, 4, 5, 7, 10, 12, 53prdsidlem 18634 . . 3 (𝜑 → ((0g𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎)))
5554simpld 495 . 2 (𝜑 → (0g𝑅) ∈ (Base‘𝑌))
5654simprd 496 . . . 4 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
5756r19.21bi 3247 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
5857simpld 495 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((0g𝑅)(+g𝑌)𝑎) = 𝑎)
5957simprd 496 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)(0g𝑅)) = 𝑎)
601, 2, 17, 52, 55, 58, 59ismndd 18624 1 (𝜑𝑌 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  Vcvv 3473  cmpt 5224  ccom 5673   Fn wfn 6527  wf 6528  cfv 6532  (class class class)co 7393  Basecbs 17126  +gcplusg 17179  0gc0g 17367  Xscprds 17373  Mndcmnd 18602
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-z 12541  df-dec 12660  df-uz 12805  df-fz 13467  df-struct 17062  df-slot 17097  df-ndx 17109  df-base 17127  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-hom 17203  df-cco 17204  df-0g 17369  df-prds 17375  df-mgm 18543  df-sgrp 18592  df-mnd 18603
This theorem is referenced by:  prds0g  18636  pwsmnd  18637  xpsmnd  18642  prdspjmhm  18685  prdsgrpd  18907  prdscmnd  19689  prdsringd  20089  dsmm0cl  21228  prdstmdd  23557
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