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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 7138 | . . 3 ⊢ ((1st ‘𝑋)𝑆(2nd ‘𝑋)) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
2 | dprd2d.1 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐴) | |
3 | 1st2nd 7720 | . . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
4 | 2, 3 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) |
5 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
6 | 4, 5 | eqeltrrd 2891 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) |
7 | df-br 5031 | . . . . . 6 ⊢ ((1st ‘𝑋)𝐴(2nd ‘𝑋) ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) | |
8 | 6, 7 | sylibr 237 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋)𝐴(2nd ‘𝑋)) |
9 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴) |
10 | elrelimasn 5920 | . . . . . 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 260 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)})) |
13 | oveq2 7143 | . . . . 5 ⊢ (𝑗 = (2nd ‘𝑋) → ((1st ‘𝑋)𝑆𝑗) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) | |
14 | eqid 2798 | . . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) | |
15 | ovex 7168 | . . . . 5 ⊢ ((1st ‘𝑋)𝑆𝑗) ∈ V | |
16 | 13, 14, 15 | fvmpt3i 6750 | . . . 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 6649 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉)) |
19 | 1, 17, 18 | 3eqtr4a 2859 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = (𝑆‘𝑋)) |
20 | sneq 4535 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → {𝑖} = {(1st ‘𝑋)}) | |
21 | 20 | imaeq2d 5896 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑋)})) |
22 | oveq1 7142 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝑖𝑆𝑗) = ((1st ‘𝑋)𝑆𝑗)) | |
23 | 21, 22 | mpteq12dv 5115 | . . . . 5 ⊢ (𝑖 = (1st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
24 | 23 | breq2d 5042 | . . . 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 3149 | . . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) |
27 | 26 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) |
28 | dprd2d.3 | . . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) | |
29 | 28 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼) |
30 | 1stdm 7721 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) | |
31 | 2, 30 | sylan 583 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) |
32 | 29, 31 | sseldd 3916 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ 𝐼) |
33 | 24, 27, 32 | rspcdva 3573 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
34 | 15, 14 | dmmpti 6464 | . . . 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 19140 | . 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 3954 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 {csn 4525 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 “ cima 5522 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 mrClscmrc 16846 SubGrpcsubg 18265 DProd cdprd 19108 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-dprd 19110 |
This theorem is referenced by: dprd2dlem1 19156 dprd2da 19157 |
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