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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 7225 | . . 3 ⊢ ((1st ‘𝑋)𝑆(2nd ‘𝑋)) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
2 | dprd2d.1 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐴) | |
3 | 1st2nd 7819 | . . . . . . . 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 2840 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) |
7 | df-br 5063 | . . . . . 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 5962 | . . . . . 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 7230 | . . . . 5 ⊢ (𝑗 = (2nd ‘𝑋) → ((1st ‘𝑋)𝑆𝑗) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) | |
14 | eqid 2738 | . . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) | |
15 | ovex 7255 | . . . . 5 ⊢ ((1st ‘𝑋)𝑆𝑗) ∈ V | |
16 | 13, 14, 15 | fvmpt3i 6832 | . . . 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 6730 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉)) |
19 | 1, 17, 18 | 3eqtr4a 2805 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = (𝑆‘𝑋)) |
20 | sneq 4560 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → {𝑖} = {(1st ‘𝑋)}) | |
21 | 20 | imaeq2d 5938 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑋)})) |
22 | oveq1 7229 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝑖𝑆𝑗) = ((1st ‘𝑋)𝑆𝑗)) | |
23 | 21, 22 | mpteq12dv 5149 | . . . . 5 ⊢ (𝑖 = (1st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
24 | 23 | breq2d 5074 | . . . 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 3106 | . . . . 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 7820 | . . . . . 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 3911 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ 𝐼) |
33 | 24, 27, 32 | rspcdva 3546 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
34 | 15, 14 | dmmpti 6531 | . . . 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 19425 | . 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 3949 | 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 1543 ∈ wcel 2111 ∀wral 3062 ⊆ wss 3875 {csn 4550 〈cop 4556 class class class wbr 5062 ↦ cmpt 5144 dom cdm 5560 “ cima 5563 Rel wrel 5565 ⟶wf 6385 ‘cfv 6389 (class class class)co 7222 1st c1st 7768 2nd c2nd 7769 mrClscmrc 17099 SubGrpcsubg 18550 DProd cdprd 19393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-iin 4916 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-se 5519 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-isom 6398 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-1st 7770 df-2nd 7771 df-supp 7913 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-er 8400 df-ixp 8588 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-fsupp 8999 df-oi 9139 df-card 9568 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-nn 11844 df-2 11906 df-n0 12104 df-z 12190 df-uz 12452 df-fz 13109 df-fzo 13252 df-seq 13588 df-hash 13910 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ress 16798 df-plusg 16828 df-0g 16959 df-gsum 16960 df-mre 17102 df-mrc 17103 df-acs 17105 df-mgm 18127 df-sgrp 18176 df-mnd 18187 df-submnd 18232 df-grp 18381 df-mulg 18502 df-subg 18553 df-cntz 18724 df-cmn 19185 df-dprd 19395 |
This theorem is referenced by: dprd2dlem1 19441 dprd2da 19442 |
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