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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 7434 | . . 3 ⊢ ((1st ‘𝑋)𝑆(2nd ‘𝑋)) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
2 | dprd2d.1 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐴) | |
3 | 1st2nd 8063 | . . . . . . . 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 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
6 | 4, 5 | eqeltrrd 2840 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) |
7 | df-br 5149 | . . . . . 6 ⊢ ((1st ‘𝑋)𝐴(2nd ‘𝑋) ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐴) | |
8 | 6, 7 | sylibr 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋)𝐴(2nd ‘𝑋)) |
9 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → Rel 𝐴) |
10 | elrelimasn 6106 | . . . . . 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 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (2nd ‘𝑋) ∈ (𝐴 “ {(1st ‘𝑋)})) |
13 | oveq2 7439 | . . . . 5 ⊢ (𝑗 = (2nd ‘𝑋) → ((1st ‘𝑋)𝑆𝑗) = ((1st ‘𝑋)𝑆(2nd ‘𝑋))) | |
14 | eqid 2735 | . . . . 5 ⊢ (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)) | |
15 | ovex 7464 | . . . . 5 ⊢ ((1st ‘𝑋)𝑆𝑗) ∈ V | |
16 | 13, 14, 15 | fvmpt3i 7021 | . . . 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 6911 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) = (𝑆‘〈(1st ‘𝑋), (2nd ‘𝑋)〉)) |
19 | 1, 17, 18 | 3eqtr4a 2801 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))‘(2nd ‘𝑋)) = (𝑆‘𝑋)) |
20 | sneq 4641 | . . . . . . 7 ⊢ (𝑖 = (1st ‘𝑋) → {𝑖} = {(1st ‘𝑋)}) | |
21 | 20 | imaeq2d 6080 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st ‘𝑋)})) |
22 | oveq1 7438 | . . . . . 6 ⊢ (𝑖 = (1st ‘𝑋) → (𝑖𝑆𝑗) = ((1st ‘𝑋)𝑆𝑗)) | |
23 | 21, 22 | mpteq12dv 5239 | . . . . 5 ⊢ (𝑖 = (1st ‘𝑋) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
24 | 23 | breq2d 5160 | . . . 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 3144 | . . . . 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 8064 | . . . . . 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 3996 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (1st ‘𝑋) ∈ 𝐼) |
33 | 24, 27, 32 | rspcdva 3623 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗))) |
34 | 15, 14 | dmmpti 6713 | . . . 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 20060 | . 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 4035 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 {csn 4631 〈cop 4637 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 “ cima 5692 Rel wrel 5694 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 mrClscmrc 17628 SubGrpcsubg 19151 DProd cdprd 20028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-dprd 20030 |
This theorem is referenced by: dprd2dlem1 20076 dprd2da 20077 |
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