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Theorem prdssgrpd 18759
Description: The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
Hypotheses
Ref Expression
prdssgrpd.y 𝑌 = (𝑆Xs𝑅)
prdssgrpd.i (𝜑𝐼𝑊)
prdssgrpd.s (𝜑𝑆𝑉)
prdssgrpd.r (𝜑𝑅:𝐼⟶Smgrp)
Assertion
Ref Expression
prdssgrpd (𝜑𝑌 ∈ Smgrp)

Proof of Theorem prdssgrpd
Dummy variables 𝑦 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2736 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2736 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdssgrpd.y . . . 4 𝑌 = (𝑆Xs𝑅)
4 eqid 2735 . . . 4 (Base‘𝑌) = (Base‘𝑌)
5 eqid 2735 . . . 4 (+g𝑌) = (+g𝑌)
6 prdssgrpd.s . . . . . 6 (𝜑𝑆𝑉)
76elexd 3502 . . . . 5 (𝜑𝑆 ∈ V)
87adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
9 prdssgrpd.i . . . . . 6 (𝜑𝐼𝑊)
109elexd 3502 . . . . 5 (𝜑𝐼 ∈ V)
1110adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
12 prdssgrpd.r . . . . 5 (𝜑𝑅:𝐼⟶Smgrp)
1312adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
14 simprl 771 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
15 simprr 773 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
163, 4, 5, 8, 11, 13, 14, 15prdsplusgsgrpcl 18758 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
17163impb 1114 . 2 ((𝜑𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
1812ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ Smgrp)
1918adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ Smgrp)
207ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ V)
2110ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
2212ffnd 6738 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
2322ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
24 simplr1 1214 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
25 simpr 484 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
263, 4, 20, 21, 23, 24, 25prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
27 simplr2 1215 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
283, 4, 20, 21, 23, 27, 25prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
29 simplr3 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
303, 4, 20, 21, 23, 29, 25prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
31 eqid 2735 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
32 eqid 2735 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
3331, 32sgrpass 18751 . . . . . 6 (((𝑅𝑦) ∈ Smgrp ∧ ((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 1371 . . . . 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 17521 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑌)𝑏)‘𝑦) = ((𝑎𝑦)(+g‘(𝑅𝑦))(𝑏𝑦)))
3635oveq1d 7446 . . . . 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 17521 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
3837oveq2d 7447 . . . . 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 2785 . . . 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 5248 . . 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 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V)
4210adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
4322adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
44163adantr3 1170 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑌)𝑏) ∈ (Base‘𝑌))
45 simpr3 1195 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
463, 4, 41, 42, 43, 44, 45, 5prdsplusgval 17520 . . 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 1193 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌))
4812adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Smgrp)
49 simpr2 1194 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
503, 4, 5, 41, 42, 48, 49, 45prdsplusgsgrpcl 18758 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
513, 4, 41, 42, 43, 47, 50, 5prdsplusgval 17520 . . 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 2785 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑌)𝑏)(+g𝑌)𝑐) = (𝑎(+g𝑌)(𝑏(+g𝑌)𝑐)))
533ovexi 7465 . . 3 𝑌 ∈ V
5453a1i 11 . 2 (𝜑𝑌 ∈ V)
551, 2, 17, 52, 54issgrpd 18756 1 (𝜑𝑌 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cmpt 5231   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Xscprds 17492  Smgrpcsgrp 18744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-prds 17494  df-mgm 18666  df-sgrp 18745
This theorem is referenced by:  prdsrngd  20194
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