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Theorem prdsmndd 17370
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 2652 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2652 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdsmndd.y . . . 4 𝑌 = (𝑆Xs𝑅)
4 eqid 2651 . . . 4 (Base‘𝑌) = (Base‘𝑌)
5 eqid 2651 . . . 4 (+g𝑌) = (+g𝑌)
6 prdsmndd.s . . . . . 6 (𝜑𝑆𝑉)
7 elex 3243 . . . . . 6 (𝑆𝑉𝑆 ∈ V)
86, 7syl 17 . . . . 5 (𝜑𝑆 ∈ V)
98adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
10 prdsmndd.i . . . . . 6 (𝜑𝐼𝑊)
11 elex 3243 . . . . . 6 (𝐼𝑊𝐼 ∈ V)
1210, 11syl 17 . . . . 5 (𝜑𝐼 ∈ V)
1312adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
14 prdsmndd.r . . . . 5 (𝜑𝑅:𝐼⟶Mnd)
1514adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
16 simprl 809 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
17 simprr 811 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
183, 4, 5, 9, 13, 15, 16, 17prdsplusgcl 17368 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
19183impb 1279 . 2 ((𝜑𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
2014ffvelrnda 6399 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
2120adantlr 751 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
228ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ V)
2312ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
24 ffn 6083 . . . . . . . . 9 (𝑅:𝐼⟶Mnd → 𝑅 Fn 𝐼)
2514, 24syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
2625ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
27 simplr1 1123 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
28 simpr 476 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
293, 4, 22, 23, 26, 27, 28prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
30 simplr2 1124 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
313, 4, 22, 23, 26, 30, 28prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
32 simplr3 1125 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
333, 4, 22, 23, 26, 32, 28prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
34 eqid 2651 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
35 eqid 2651 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
3634, 35mndass 17349 . . . . . 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))))
3721, 29, 31, 33, 36syl13anc 1368 . . . . 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))))
383, 4, 22, 23, 26, 27, 30, 5, 28prdsplusgfval 16181 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑌)𝑏)‘𝑦) = ((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦)))
3938oveq1d 6705 . . . . 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)))
403, 4, 22, 23, 26, 30, 32, 5, 28prdsplusgfval 16181 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
4140oveq2d 6706 . . . . 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))))
4237, 39, 413eqtr4d 2695 . . . 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)))
4342mpteq2dva 4777 . . 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))))
448adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
4512adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
4625adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
47183adantr3 1242 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
48 simpr3 1089 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
493, 4, 44, 45, 46, 47, 48, 5prdsplusgval 16180 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑦𝐼 ↦ (((𝑎(+g𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
50 simpr1 1087 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
5114adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
52 simpr2 1088 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
533, 4, 5, 44, 45, 51, 52, 48prdsplusgcl 17368 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
543, 4, 44, 45, 46, 50, 53, 5prdsplusgval 16180 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)) = (𝑦𝐼 ↦ ((𝑎𝑦)(+g‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
5543, 49, 543eqtr4d 2695 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)))
56 eqid 2651 . . . 4 (0g𝑅) = (0g𝑅)
573, 4, 5, 8, 12, 14, 56prdsidlem 17369 . . 3 (𝜑 → ((0g𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎)))
5857simpld 474 . 2 (𝜑 → (0g𝑅) ∈ (Base‘𝑌))
5957simprd 478 . . . 4 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
6059r19.21bi 2961 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (((0g𝑅)(+g𝑌)𝑎) = 𝑎 ∧ (𝑎(+g𝑌)(0g𝑅)) = 𝑎))
6160simpld 474 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((0g𝑅)(+g𝑌)𝑎) = 𝑎)
6260simprd 478 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)(0g𝑅)) = 𝑎)
631, 2, 19, 55, 58, 61, 62ismndd 17360 1 (𝜑𝑌 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cmpt 4762  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Xscprds 16153  Mndcmnd 17341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-prds 16155  df-mgm 17289  df-sgrp 17331  df-mnd 17342
This theorem is referenced by:  prds0g  17371  pwsmnd  17372  xpsmnd  17377  prdspjmhm  17414  prdsgrpd  17572  prdscmnd  18310  prdsringd  18658  dsmm0cl  20132  prdstmdd  21974
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