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Theorem dprd2dlem2 18794
 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 6909 . . 3 ((1st𝑋)𝑆(2nd𝑋)) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩)
2 dprd2d.1 . . . . . . . 8 (𝜑 → Rel 𝐴)
3 1st2nd 7477 . . . . . . . 8 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
42, 3sylan 577 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 simpr 479 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋𝐴)
64, 5eqeltrrd 2908 . . . . . 6 ((𝜑𝑋𝐴) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
7 df-br 4875 . . . . . 6 ((1st𝑋)𝐴(2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
86, 7sylibr 226 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋)𝐴(2nd𝑋))
92adantr 474 . . . . . 6 ((𝜑𝑋𝐴) → Rel 𝐴)
10 elrelimasn 5731 . . . . . 6 (Rel 𝐴 → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
119, 10syl 17 . . . . 5 ((𝜑𝑋𝐴) → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
128, 11mpbird 249 . . . 4 ((𝜑𝑋𝐴) → (2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}))
13 oveq2 6914 . . . . 5 (𝑗 = (2nd𝑋) → ((1st𝑋)𝑆𝑗) = ((1st𝑋)𝑆(2nd𝑋)))
14 eqid 2826 . . . . 5 (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))
15 ovex 6938 . . . . 5 ((1st𝑋)𝑆𝑗) ∈ V
1613, 14, 15fvmpt3i 6535 . . . 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 6438 . . 3 ((𝜑𝑋𝐴) → (𝑆𝑋) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩))
191, 17, 183eqtr4a 2888 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = (𝑆𝑋))
20 sneq 4408 . . . . . . 7 (𝑖 = (1st𝑋) → {𝑖} = {(1st𝑋)})
2120imaeq2d 5708 . . . . . 6 (𝑖 = (1st𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑋)}))
22 oveq1 6913 . . . . . 6 (𝑖 = (1st𝑋) → (𝑖𝑆𝑗) = ((1st𝑋)𝑆𝑗))
2321, 22mpteq12dv 4957 . . . . 5 (𝑖 = (1st𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
2423breq2d 4886 . . . 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 3176 . . . . 5 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2726adantr 474 . . . 4 ((𝜑𝑋𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28 dprd2d.3 . . . . . 6 (𝜑 → dom 𝐴𝐼)
2928adantr 474 . . . . 5 ((𝜑𝑋𝐴) → dom 𝐴𝐼)
30 1stdm 7478 . . . . . 6 ((Rel 𝐴𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
312, 30sylan 577 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
3229, 31sseldd 3829 . . . 4 ((𝜑𝑋𝐴) → (1st𝑋) ∈ 𝐼)
3324, 27, 32rspcdva 3533 . . 3 ((𝜑𝑋𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
3415, 14dmmpti 6257 . . . 4 dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)})
3534a1i 11 . . 3 ((𝜑𝑋𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)}))
3633, 35, 12dprdub 18779 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3719, 36eqsstr3d 3866 1 ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118   ⊆ wss 3799  {csn 4398  ⟨cop 4404   class class class wbr 4874   ↦ cmpt 4953  dom cdm 5343   “ cima 5346  Rel wrel 5348  ⟶wf 6120  ‘cfv 6124  (class class class)co 6906  1st c1st 7427  2nd c2nd 7428  mrClscmrc 16597  SubGrpcsubg 17940   DProd cdprd 18747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-inf2 8816  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-iin 4744  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-supp 7561  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-ixp 8177  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-fsupp 8546  df-oi 8685  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-seq 13097  df-hash 13412  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-plusg 16319  df-0g 16456  df-gsum 16457  df-mre 16600  df-mrc 16601  df-acs 16603  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-submnd 17690  df-grp 17780  df-mulg 17896  df-subg 17943  df-cntz 18101  df-cmn 18549  df-dprd 18749 This theorem is referenced by:  dprd2dlem1  18795  dprd2da  18796
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