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Theorem dprd2dlem2 18706
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 6845 . . 3 ((1st𝑋)𝑆(2nd𝑋)) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩)
2 dprd2d.1 . . . . . . . 8 (𝜑 → Rel 𝐴)
3 1st2nd 7414 . . . . . . . 8 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
42, 3sylan 575 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 simpr 477 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋𝐴)
64, 5eqeltrrd 2845 . . . . . 6 ((𝜑𝑋𝐴) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
7 df-br 4810 . . . . . 6 ((1st𝑋)𝐴(2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
86, 7sylibr 225 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋)𝐴(2nd𝑋))
92adantr 472 . . . . . 6 ((𝜑𝑋𝐴) → Rel 𝐴)
10 elrelimasn 5671 . . . . . 6 (Rel 𝐴 → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
119, 10syl 17 . . . . 5 ((𝜑𝑋𝐴) → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
128, 11mpbird 248 . . . 4 ((𝜑𝑋𝐴) → (2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}))
13 oveq2 6850 . . . . 5 (𝑗 = (2nd𝑋) → ((1st𝑋)𝑆𝑗) = ((1st𝑋)𝑆(2nd𝑋)))
14 eqid 2765 . . . . 5 (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))
15 ovex 6874 . . . . 5 ((1st𝑋)𝑆𝑗) ∈ V
1613, 14, 15fvmpt3i 6476 . . . 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 6379 . . 3 ((𝜑𝑋𝐴) → (𝑆𝑋) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩))
191, 17, 183eqtr4a 2825 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = (𝑆𝑋))
20 sneq 4344 . . . . . . 7 (𝑖 = (1st𝑋) → {𝑖} = {(1st𝑋)})
2120imaeq2d 5648 . . . . . 6 (𝑖 = (1st𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑋)}))
22 oveq1 6849 . . . . . 6 (𝑖 = (1st𝑋) → (𝑖𝑆𝑗) = ((1st𝑋)𝑆𝑗))
2321, 22mpteq12dv 4892 . . . . 5 (𝑖 = (1st𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
2423breq2d 4821 . . . 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 3113 . . . . 5 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2726adantr 472 . . . 4 ((𝜑𝑋𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28 dprd2d.3 . . . . . 6 (𝜑 → dom 𝐴𝐼)
2928adantr 472 . . . . 5 ((𝜑𝑋𝐴) → dom 𝐴𝐼)
30 1stdm 7415 . . . . . 6 ((Rel 𝐴𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
312, 30sylan 575 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
3229, 31sseldd 3762 . . . 4 ((𝜑𝑋𝐴) → (1st𝑋) ∈ 𝐼)
3324, 27, 32rspcdva 3467 . . 3 ((𝜑𝑋𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
3415, 14dmmpti 6201 . . . 4 dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)})
3534a1i 11 . . 3 ((𝜑𝑋𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)}))
3633, 35, 12dprdub 18691 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3719, 36eqsstr3d 3800 1 ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732  {csn 4334  cop 4340   class class class wbr 4809  cmpt 4888  dom cdm 5277  cima 5280  Rel wrel 5282  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  mrClscmrc 16511  SubGrpcsubg 17854   DProd cdprd 18659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-0g 16370  df-gsum 16371  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-submnd 17604  df-grp 17694  df-mulg 17810  df-subg 17857  df-cntz 18015  df-cmn 18461  df-dprd 18661
This theorem is referenced by:  dprd2dlem1  18707  dprd2da  18708
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