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Theorem dprd2dlem2 18360
 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 6607 . . 3 ((1st𝑋)𝑆(2nd𝑋)) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩)
2 dprd2d.1 . . . . . . . 8 (𝜑 → Rel 𝐴)
3 1st2nd 7159 . . . . . . . 8 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
42, 3sylan 488 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 simpr 477 . . . . . . 7 ((𝜑𝑋𝐴) → 𝑋𝐴)
64, 5eqeltrrd 2699 . . . . . 6 ((𝜑𝑋𝐴) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
7 df-br 4614 . . . . . 6 ((1st𝑋)𝐴(2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴)
86, 7sylibr 224 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋)𝐴(2nd𝑋))
92adantr 481 . . . . . 6 ((𝜑𝑋𝐴) → Rel 𝐴)
10 elrelimasn 5448 . . . . . 6 (Rel 𝐴 → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
119, 10syl 17 . . . . 5 ((𝜑𝑋𝐴) → ((2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}) ↔ (1st𝑋)𝐴(2nd𝑋)))
128, 11mpbird 247 . . . 4 ((𝜑𝑋𝐴) → (2nd𝑋) ∈ (𝐴 “ {(1st𝑋)}))
13 oveq2 6612 . . . . 5 (𝑗 = (2nd𝑋) → ((1st𝑋)𝑆𝑗) = ((1st𝑋)𝑆(2nd𝑋)))
14 eqid 2621 . . . . 5 (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))
15 ovex 6632 . . . . 5 ((1st𝑋)𝑆𝑗) ∈ V
1613, 14, 15fvmpt3i 6244 . . . 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 6152 . . 3 ((𝜑𝑋𝐴) → (𝑆𝑋) = (𝑆‘⟨(1st𝑋), (2nd𝑋)⟩))
191, 17, 183eqtr4a 2681 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) = (𝑆𝑋))
20 dprd2d.3 . . . . . 6 (𝜑 → dom 𝐴𝐼)
2120adantr 481 . . . . 5 ((𝜑𝑋𝐴) → dom 𝐴𝐼)
22 1stdm 7160 . . . . . 6 ((Rel 𝐴𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
232, 22sylan 488 . . . . 5 ((𝜑𝑋𝐴) → (1st𝑋) ∈ dom 𝐴)
2421, 23sseldd 3584 . . . 4 ((𝜑𝑋𝐴) → (1st𝑋) ∈ 𝐼)
25 dprd2d.4 . . . . . 6 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2625ralrimiva 2960 . . . . 5 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
2726adantr 481 . . . 4 ((𝜑𝑋𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28 sneq 4158 . . . . . . . 8 (𝑖 = (1st𝑋) → {𝑖} = {(1st𝑋)})
2928imaeq2d 5425 . . . . . . 7 (𝑖 = (1st𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑋)}))
30 oveq1 6611 . . . . . . 7 (𝑖 = (1st𝑋) → (𝑖𝑆𝑗) = ((1st𝑋)𝑆𝑗))
3129, 30mpteq12dv 4693 . . . . . 6 (𝑖 = (1st𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
3231breq2d 4625 . . . . 5 (𝑖 = (1st𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3332rspcv 3291 . . . 4 ((1st𝑋) ∈ 𝐼 → (∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3424, 27, 33sylc 65 . . 3 ((𝜑𝑋𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)))
3515, 14dmmpti 5980 . . . 4 dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)})
3635a1i 11 . . 3 ((𝜑𝑋𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗)) = (𝐴 “ {(1st𝑋)}))
3734, 36, 12dprdub 18345 . 2 ((𝜑𝑋𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))‘(2nd𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
3819, 37eqsstr3d 3619 1 ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ⊆ wss 3555  {csn 4148  ⟨cop 4154   class class class wbr 4613   ↦ cmpt 4673  dom cdm 5074   “ cima 5077  Rel wrel 5079  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  mrClscmrc 16164  SubGrpcsubg 17509   DProd cdprd 18313 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-mulg 17462  df-subg 17512  df-cntz 17671  df-cmn 18116  df-dprd 18315 This theorem is referenced by:  dprd2dlem1  18361  dprd2da  18362
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