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Mirrors > Home > MPE Home > Th. List > dprdf1 | Structured version Visualization version GIF version |
Description: Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdf1.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprdf1.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dprdf1.3 | ⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼) |
Ref | Expression |
---|---|
dprdf1 | ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdf1.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dprdf1.2 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | dprdf1.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼) | |
4 | f1f 6101 | . . . . . . . 8 ⊢ (𝐹:𝐽–1-1→𝐼 → 𝐹:𝐽⟶𝐼) | |
5 | frn 6053 | . . . . . . . 8 ⊢ (𝐹:𝐽⟶𝐼 → ran 𝐹 ⊆ 𝐼) | |
6 | 3, 4, 5 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐼) |
7 | 1, 2, 6 | dprdres 18427 | . . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ ran 𝐹) ∧ (𝐺 DProd (𝑆 ↾ ran 𝐹)) ⊆ (𝐺 DProd 𝑆))) |
8 | 7 | simpld 475 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ ran 𝐹)) |
9 | 1, 2 | dprdf2 18406 | . . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
10 | 9, 6 | fssresd 6071 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾ ran 𝐹):ran 𝐹⟶(SubGrp‘𝐺)) |
11 | fdm 6051 | . . . . . 6 ⊢ ((𝑆 ↾ ran 𝐹):ran 𝐹⟶(SubGrp‘𝐺) → dom (𝑆 ↾ ran 𝐹) = ran 𝐹) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ ran 𝐹) = ran 𝐹) |
13 | f1f1orn 6148 | . . . . . 6 ⊢ (𝐹:𝐽–1-1→𝐼 → 𝐹:𝐽–1-1-onto→ran 𝐹) | |
14 | 3, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐽–1-1-onto→ran 𝐹) |
15 | 8, 12, 14 | dprdf1o 18431 | . . . 4 ⊢ (𝜑 → (𝐺dom DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹) ∧ (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹)))) |
16 | 15 | simpld 475 | . . 3 ⊢ (𝜑 → 𝐺dom DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) |
17 | ssid 3624 | . . . 4 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
18 | cores 5638 | . . . 4 ⊢ (ran 𝐹 ⊆ ran 𝐹 → ((𝑆 ↾ ran 𝐹) ∘ 𝐹) = (𝑆 ∘ 𝐹)) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ ((𝑆 ↾ ran 𝐹) ∘ 𝐹) = (𝑆 ∘ 𝐹) |
20 | 16, 19 | syl6breq 4694 | . 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹)) |
21 | 19 | oveq2i 6661 | . . . 4 ⊢ (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ∘ 𝐹)) |
22 | 15 | simprd 479 | . . . 4 ⊢ (𝜑 → (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹))) |
23 | 21, 22 | syl5eqr 2670 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹))) |
24 | 7 | simprd 479 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ ran 𝐹)) ⊆ (𝐺 DProd 𝑆)) |
25 | 23, 24 | eqsstrd 3639 | . 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆)) |
26 | 20, 25 | jca 554 | 1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 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-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 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-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-cmn 18195 df-dprd 18394 |
This theorem is referenced by: (None) |
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