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Theorem dprd2dlem2 19983

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	⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
dprd2d.3	⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
dprd2d.4	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
dprd2d.5	⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
dprd2d.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))

**Assertion**
Ref	Expression
dprd2dlem2	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)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**
Step	Hyp	Ref	Expression
1		df-ov 7371	. . 3 ⊢ ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
2		dprd2d.1	. . . . . . . 8 ⊢ (𝜑 → Rel 𝐴)
3		1st2nd 7993	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4	2, 3	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
5		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
6	4, 5	eqeltrrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
7		df-br 5101	. . . . . 6 ⊢ ((1^st ‘𝑋)𝐴(2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴)
8	6, 7	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋)𝐴(2^nd ‘𝑋))
9	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴)
10		elrelimasn 6053	. . . . . 6 ⊢ (Rel 𝐴 → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) ↔ (1^st ‘𝑋)𝐴(2^nd ‘𝑋)))
12	8, 11	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}))
13		oveq2 7376	. . . . 5 ⊢ (𝑗 = (2^nd ‘𝑋) → ((1^st ‘𝑋)𝑆𝑗) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
14		eqid 2737	. . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))
15		ovex 7401	. . . . 5 ⊢ ((1^st ‘𝑋)𝑆𝑗) ∈ V
16	13, 14, 15	fvmpt3i 6955	. . . 4 ⊢ ((2^nd ‘𝑋) ∈ (𝐴 “ {(1^st ‘𝑋)}) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
17	12, 16	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = ((1^st ‘𝑋)𝑆(2^nd ‘𝑋)))
18	4	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
19	1, 17, 18	3eqtr4a 2798	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) = (𝑆‘𝑋))
20		sneq 4592	. . . . . . 7 ⊢ (𝑖 = (1^st ‘𝑋) → {𝑖} = {(1^st ‘𝑋)})
21	20	imaeq2d 6027	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1^st ‘𝑋)}))
22		oveq1 7375	. . . . . 6 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑖𝑆𝑗) = ((1^st ‘𝑋)𝑆𝑗))
23	21, 22	mpteq12dv 5187	. . . . 5 ⊢ (𝑖 = (1^st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
24	23	breq2d 5112	. . . 4 ⊢ (𝑖 = (1^st ‘𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))
25		dprd2d.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
26	25	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
28		dprd2d.3	. . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼)
30		1stdm 7994	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
31	2, 30	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ dom 𝐴)
32	29, 31	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1^st ‘𝑋) ∈ 𝐼)
33	24, 27, 32	rspcdva 3579	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)))
34	15, 14	dmmpti 6644	. . . 4 ⊢ dom (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1^st ‘𝑋)})
35	34	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1^st ‘𝑋)}))
36	33, 35, 12	dprdub 19968	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))‘(2^nd ‘𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))
37	19, 36	eqsstrrd 3971	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3903 {csn 4582 ⟨cop 4588 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 “ cima 5635 Rel wrel 5637 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 mrClscmrc 17514 SubGrpcsubg 19062 DProd cdprd 19936

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-0g 17373 df-gsum 17374 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-mulg 19010 df-subg 19065 df-cntz 19258 df-cmn 19723 df-dprd 19938

This theorem is referenced by: dprd2dlem1 19984 dprd2da 19985

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