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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 6574 | . . . . . . . 8 ⊢ (𝐹:𝐽–1-1→𝐼 → 𝐹:𝐽⟶𝐼) | |
5 | frn 6519 | . . . . . . . 8 ⊢ (𝐹:𝐽⟶𝐼 → ran 𝐹 ⊆ 𝐼) | |
6 | 3, 4, 5 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐼) |
7 | 1, 2, 6 | dprdres 19149 | . . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ ran 𝐹) ∧ (𝐺 DProd (𝑆 ↾ ran 𝐹)) ⊆ (𝐺 DProd 𝑆))) |
8 | 7 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ ran 𝐹)) |
9 | 1, 2 | dprdf2 19128 | . . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
10 | 9, 6 | fssresd 6544 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾ ran 𝐹):ran 𝐹⟶(SubGrp‘𝐺)) |
11 | 10 | fdmd 6522 | . . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ ran 𝐹) = ran 𝐹) |
12 | f1f1orn 6625 | . . . . . 6 ⊢ (𝐹:𝐽–1-1→𝐼 → 𝐹:𝐽–1-1-onto→ran 𝐹) | |
13 | 3, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐽–1-1-onto→ran 𝐹) |
14 | 8, 11, 13 | dprdf1o 19153 | . . . 4 ⊢ (𝜑 → (𝐺dom DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹) ∧ (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹)))) |
15 | 14 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐺dom DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) |
16 | ssid 3988 | . . . 4 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
17 | cores 6101 | . . . 4 ⊢ (ran 𝐹 ⊆ ran 𝐹 → ((𝑆 ↾ ran 𝐹) ∘ 𝐹) = (𝑆 ∘ 𝐹)) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ ((𝑆 ↾ ran 𝐹) ∘ 𝐹) = (𝑆 ∘ 𝐹) |
19 | 15, 18 | breqtrdi 5106 | . 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹)) |
20 | 18 | oveq2i 7166 | . . . 4 ⊢ (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ∘ 𝐹)) |
21 | 14 | simprd 498 | . . . 4 ⊢ (𝜑 → (𝐺 DProd ((𝑆 ↾ ran 𝐹) ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹))) |
22 | 20, 21 | syl5eqr 2870 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd (𝑆 ↾ ran 𝐹))) |
23 | 7 | simprd 498 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ ran 𝐹)) ⊆ (𝐺 DProd 𝑆)) |
24 | 22, 23 | eqsstrd 4004 | . 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆)) |
25 | 19, 24 | jca 514 | 1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ⊆ wss 3935 class class class wbr 5065 dom cdm 5554 ran crn 5555 ↾ cres 5556 ∘ ccom 5558 ⟶wf 6350 –1-1→wf1 6351 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 SubGrpcsubg 18272 DProd cdprd 19114 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-gim 18398 df-cntz 18446 df-oppg 18473 df-cmn 18907 df-dprd 19116 |
This theorem is referenced by: (None) |
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