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Theorem dprd2dlem2 19162
Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dprd2d.1 (𝜑 → Rel 𝐴)
dprd2d.2 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
dprd2d.3 (𝜑 → dom 𝐴𝐼)
dprd2d.4 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
dprd2d.5 (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
dprd2d.k 𝐾 = (mrCls‘(SubGrp‘𝐺))
Assertion
Ref Expression
dprd2dlem2 ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
Distinct variable groups:   𝑖,𝑗,𝐴   𝑖,𝐺,𝑗   𝑖,𝐼   𝑖,𝐾   𝜑,𝑖,𝑗   𝑆,𝑖,𝑗   𝑖,𝑋,𝑗
Allowed substitution hints:   𝐼(𝑗)   𝐾(𝑗)

Proof of Theorem dprd2dlem2
StepHypRef Expression
1 df-ov 7152 . . 3 ((1st𝑋)𝑆(2nd𝑋)) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩)
2 dprd2d.1 . . . . . . . 8 (𝜑 → Rel 𝐴)
3 1st2nd 7733 . . . . . . . 8 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
42, 3sylan 583 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 simpr 488 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋𝐴)
64, 5eqeltrrd 2917 . . . . . 6 ((𝜑𝑋𝐴) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
7 df-br 5053 . . . . . 6 ((1st𝑋)𝐴(2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
86, 7sylibr 237 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋)𝐴(2nd𝑋))
92adantr 484 . . . . . 6 ((𝜑𝑋𝐴) → Rel 𝐴)
10 elrelimasn 5940 . . . . . 6 (Rel 𝐴 → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
119, 10syl 17 . . . . 5 ((𝜑𝑋𝐴) → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
128, 11mpbird 260 . . . 4 ((𝜑𝑋𝐴) → (2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}))
13 oveq2 7157 . . . . 5 (𝑗 = (2nd𝑋) → ((1st𝑋)𝑆𝑗) = ((1st𝑋)𝑆(2nd𝑋)))
14 eqid 2824 . . . . 5 (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))
15 ovex 7182 . . . . 5 ((1st𝑋)𝑆𝑗) ∈ V
1613, 14, 15fvmpt3i 6764 . . . 4 ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = ((1st𝑋)𝑆(2nd𝑋)))
1712, 16syl 17 . . 3 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = ((1st𝑋)𝑆(2nd𝑋)))
184fveq2d 6665 . . 3 ((𝜑𝑋𝐴) → (𝑆𝑋) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩))
191, 17, 183eqtr4a 2885 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = (𝑆𝑋))
20 sneq 4560 . . . . . . 7 (𝑖 = (1st𝑋) → {𝑖} = {(1st𝑋)})
2120imaeq2d 5916 . . . . . 6 (𝑖 = (1st𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑋)}))
22 oveq1 7156 . . . . . 6 (𝑖 = (1st𝑋) → (𝑖𝑆𝑗) = ((1st𝑋)𝑆𝑗))
2321, 22mpteq12dv 5137 . . . . 5 (𝑖 = (1st𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
2423breq2d 5064 . . . 4 (𝑖 = (1st𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
25 dprd2d.4 . . . . . 6 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2625ralrimiva 3177 . . . . 5 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2726adantr 484 . . . 4 ((𝜑𝑋𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28 dprd2d.3 . . . . . 6 (𝜑 → dom 𝐴𝐼)
2928adantr 484 . . . . 5 ((𝜑𝑋𝐴) → dom 𝐴𝐼)
30 1stdm 7734 . . . . . 6 ((Rel 𝐴𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
312, 30sylan 583 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
3229, 31sseldd 3954 . . . 4 ((𝜑𝑋𝐴) → (1st𝑋) ∈ 𝐼)
3324, 27, 32rspcdva 3611 . . 3 ((𝜑𝑋𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
3415, 14dmmpti 6481 . . . 4 dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)})
3534a1i 11 . . 3 ((𝜑𝑋𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)}))
3633, 35, 12dprdub 19147 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3719, 36eqsstrrd 3992 1 ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  {csn 4550  cop 4556   class class class wbr 5052  cmpt 5132  dom cdm 5542  cima 5545  Rel wrel 5547  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  mrClscmrc 16854  SubGrpcsubg 18273   DProd cdprd 19115
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-mulg 18225  df-subg 18276  df-cntz 18447  df-cmn 18908  df-dprd 19117
This theorem is referenced by:  dprd2dlem1  19163  dprd2da  19164
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