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Mirrors > Home > MPE Home > Th. List > dpjlsm | Structured version Visualization version GIF version |
Description: The two subgroups that appear in dpjval 19172 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjlem.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
dpjlsm.s | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
dpjlsm | ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpjfval.1 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dpjfval.2 | . . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | 1, 2 | dprdf2 19123 | . . 3 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
4 | disjdif 4420 | . . . 4 ⊢ ({𝑋} ∩ (𝐼 ∖ {𝑋})) = ∅ | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝑋} ∩ (𝐼 ∖ {𝑋})) = ∅) |
6 | undif2 4424 | . . . 4 ⊢ ({𝑋} ∪ (𝐼 ∖ {𝑋})) = ({𝑋} ∪ 𝐼) | |
7 | dpjlem.3 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
8 | 7 | snssd 4735 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐼) |
9 | ssequn1 4155 | . . . . 5 ⊢ ({𝑋} ⊆ 𝐼 ↔ ({𝑋} ∪ 𝐼) = 𝐼) | |
10 | 8, 9 | sylib 220 | . . . 4 ⊢ (𝜑 → ({𝑋} ∪ 𝐼) = 𝐼) |
11 | 6, 10 | syl5req 2869 | . . 3 ⊢ (𝜑 → 𝐼 = ({𝑋} ∪ (𝐼 ∖ {𝑋}))) |
12 | dpjlsm.s | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
13 | 3, 5, 11, 12, 1 | dprdsplit 19164 | . 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ {𝑋})) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
14 | 1, 2, 7 | dpjlem 19167 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ {𝑋})) = (𝑆‘𝑋)) |
15 | 14 | oveq1d 7165 | . 2 ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ {𝑋})) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = ((𝑆‘𝑋) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
16 | 13, 15 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 LSSumclsm 18753 DProd cdprd 19109 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-dprd 19111 |
This theorem is referenced by: dpjf 19173 dpjidcl 19174 dpjghm 19179 |
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