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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 6653 | . . 3 ⊢ ((1st ‘𝑋)𝑆(2nd ‘𝑋)) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
2 | dprd2d.1 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐴) | |
3 | 1st2nd 7214 | . . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
4 | 2, 3 | sylan 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) |
5 | simpr 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
6 | 4, 5 | eqeltrrd 2702 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) |
7 | df-br 4654 | . . . . . 6 ⊢ ((1st ‘𝑋)𝐴(2nd ‘𝑋) ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) | |
8 | 6, 7 | sylibr 224 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋)𝐴(2nd ‘𝑋)) |
9 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴) |
10 | elrelimasn 5489 | . . . . . 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 247 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)})) |
13 | oveq2 6658 | . . . . 5 ⊢ (𝑗 = (2nd ‘𝑋) → ((1st ‘𝑋)𝑆𝑗) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) | |
14 | eqid 2622 | . . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) | |
15 | ovex 6678 | . . . . 5 ⊢ ((1st ‘𝑋)𝑆𝑗) ∈ V | |
16 | 13, 14, 15 | fvmpt3i 6287 | . . . 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 6195 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉)) |
19 | 1, 17, 18 | 3eqtr4a 2682 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = (𝑆‘𝑋)) |
20 | dprd2d.3 | . . . . . 6 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) | |
21 | 20 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom 𝐴 ⊆ 𝐼) |
22 | 1stdm 7215 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) | |
23 | 2, 22 | sylan 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ dom 𝐴) |
24 | 21, 23 | sseldd 3604 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ 𝐼) |
25 | dprd2d.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) | |
26 | 25 | ralrimiva 2966 | . . . . 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 | sneq 4187 | . . . . . . . 8 ⊢ (𝑖 = (1st ‘𝑋) → {𝑖} = {(1st ‘𝑋)}) | |
29 | 28 | imaeq2d 5466 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑋)})) |
30 | oveq1 6657 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → (𝑖𝑆𝑗) = ((1st ‘𝑋)𝑆𝑗)) | |
31 | 29, 30 | mpteq12dv 4733 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
32 | 31 | breq2d 4665 | . . . . 5 ⊢ (𝑖 = (1st ‘𝑋) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
33 | 32 | rspcv 3305 | . . . 4 ⊢ ((1st ‘𝑋) ∈ 𝐼 → (∀𝑖 ∈ 𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
34 | 24, 27, 33 | sylc 65 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
35 | 15, 14 | dmmpti 6023 | . . . 4 ⊢ dom (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1st ‘𝑋)}) |
36 | 35 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝐴 “ {(1st ‘𝑋)})) |
37 | 34, 36, 12 | dprdub 18424 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) |
38 | 19, 37 | eqsstr3d 3640 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 “ cima 5117 Rel wrel 5119 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 mrClscmrc 16243 SubGrpcsubg 17588 DProd cdprd 18392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-dprd 18394 |
This theorem is referenced by: dprd2dlem1 18440 dprd2da 18441 |
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