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Theorem prdsmndd 17146
 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 2611 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2611 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdsmndd.y . . . 4 𝑌 = (𝑆Xs𝑅)
4 eqid 2610 . . . 4 (Base‘𝑌) = (Base‘𝑌)
5 eqid 2610 . . . 4 (+g𝑌) = (+g𝑌)
6 prdsmndd.s . . . . . 6 (𝜑𝑆𝑉)
7 elex 3185 . . . . . 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 3185 . . . . . 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 790 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
17 simprr 792 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
183, 4, 5, 9, 13, 15, 16, 17prdsplusgcl 17144 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
19183impb 1252 . 2 ((𝜑𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
2014ffvelrnda 6267 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
2120adantlr 747 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ Mnd)
228ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ V)
2312ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
24 ffn 5958 . . . . . . . . 9 (𝑅:𝐼⟶Mnd → 𝑅 Fn 𝐼)
2514, 24syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
2625ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
27 simplr1 1096 . . . . . . 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 15955 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
30 simplr2 1097 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
313, 4, 22, 23, 26, 30, 28prdsbasprj 15955 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
32 simplr3 1098 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
333, 4, 22, 23, 26, 32, 28prdsbasprj 15955 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
34 eqid 2610 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
35 eqid 2610 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
3634, 35mndass 17125 . . . . . 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 1320 . . . . 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 15957 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑌)𝑏)‘𝑦) = ((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦)))
3938oveq1d 6564 . . . . 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 15957 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
4140oveq2d 6565 . . . . 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 2654 . . . 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 4672 . . 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 1215 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
48 simpr3 1062 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
493, 4, 44, 45, 46, 47, 48, 5prdsplusgval 15956 . . 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 1060 . . . 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 1061 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
533, 4, 5, 44, 45, 51, 52, 48prdsplusgcl 17144 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
543, 4, 44, 45, 46, 50, 53, 5prdsplusgval 15956 . . 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 2654 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)))
56 eqid 2610 . . . 4 (0g𝑅) = (0g𝑅)
573, 4, 5, 8, 12, 14, 56prdsidlem 17145 . . 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 2916 . . 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 17136 1 (𝜑𝑌 ∈ Mnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ↦ cmpt 4643   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Xscprds 15929  Mndcmnd 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-prds 15931  df-mgm 17065  df-sgrp 17107  df-mnd 17118 This theorem is referenced by:  prds0g  17147  pwsmnd  17148  xpsmnd  17153  prdspjmhm  17190  prdsgrpd  17348  prdscmnd  18087  prdsringd  18435  dsmm0cl  19903  prdstmdd  21737
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