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Mirrors > Home > MPE Home > Th. List > dprd2dlem2 | Structured version Visualization version GIF version |
Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.) |
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)) |
Ref | Expression |
---|---|
dprd2dlem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7148 | . . 3 ⊢ ((1st ‘𝑋)𝑆(2nd ‘𝑋)) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
2 | dprd2d.1 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐴) | |
3 | 1st2nd 7727 | . . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
4 | 2, 3 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) |
5 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
6 | 4, 5 | eqeltrrd 2911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) |
7 | df-br 5058 | . . . . . 6 ⊢ ((1st ‘𝑋)𝐴(2nd ‘𝑋) ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) | |
8 | 6, 7 | sylibr 235 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋)𝐴(2nd ‘𝑋)) |
9 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴) |
10 | elrelimasn 5946 | . . . . . 6 ⊢ (Rel 𝐴 → ((2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)}) ↔ (1st ‘𝑋)𝐴(2nd ‘𝑋))) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)}) ↔ (1st ‘𝑋)𝐴(2nd ‘𝑋))) |
12 | 8, 11 | mpbird 258 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)})) |
13 | oveq2 7153 | . . . . 5 ⊢ (𝑗 = (2nd ‘𝑋) → ((1st ‘𝑋)𝑆𝑗) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) | |
14 | eqid 2818 | . . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) | |
15 | ovex 7178 | . . . . 5 ⊢ ((1st ‘𝑋)𝑆𝑗) ∈ V | |
16 | 13, 14, 15 | fvmpt3i 6766 | . . . 4 ⊢ ((2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)}) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) |
17 | 12, 16 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) |
18 | 4 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉)) |
19 | 1, 17, 18 | 3eqtr4a 2879 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = (𝑆‘𝑋)) |
20 | sneq 4567 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → {𝑖} = {(1st ‘𝑋)}) | |
21 | 20 | imaeq2d 5922 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑋)})) |
22 | oveq1 7152 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝑖𝑆𝑗) = ((1st ‘𝑋)𝑆𝑗)) | |
23 | 21, 22 | mpteq12dv 5142 | . . . . 5 ⊢ (𝑖 = (1st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
24 | 23 | breq2d 5069 | . . . 4 ⊢ (𝑖 = (1st ‘𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
25 | dprd2d.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) | |
26 | 25 | ralrimiva 3179 | . . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) |
27 | 26 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) |
28 | dprd2d.3 | . . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) | |
29 | 28 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼) |
30 | 1stdm 7728 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) | |
31 | 2, 30 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) |
32 | 29, 31 | sseldd 3965 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ 𝐼) |
33 | 24, 27, 32 | rspcdva 3622 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
34 | 15, 14 | dmmpti 6485 | . . . 4 ⊢ dom (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1st ‘𝑋)}) |
35 | 34 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1st ‘𝑋)})) |
36 | 33, 35, 12 | dprdub 19076 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
37 | 19, 36 | eqsstrrd 4003 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 {csn 4557 〈cop 4563 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 “ cima 5551 Rel wrel 5553 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 mrClscmrc 16842 SubGrpcsubg 18211 DProd cdprd 19044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-dprd 19046 |
This theorem is referenced by: dprd2dlem1 19092 dprd2da 19093 |
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