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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 6575 | . . . . . . . 8 ⊢ (𝐹:𝐽–1-1→𝐼 → 𝐹:𝐽⟶𝐼) | |
5 | frn 6520 | . . . . . . . 8 ⊢ (𝐹:𝐽⟶𝐼 → ran 𝐹 ⊆ 𝐼) | |
6 | 3, 4, 5 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐼) |
7 | 1, 2, 6 | dprdres 19150 | . . . . . 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 19129 | . . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
10 | 9, 6 | fssresd 6545 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾ ran 𝐹):ran 𝐹⟶(SubGrp‘𝐺)) |
11 | 10 | fdmd 6523 | . . . . 5 ⊢ (𝜑 → dom (𝑆 ↾ ran 𝐹) = ran 𝐹) |
12 | f1f1orn 6626 | . . . . . 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 19154 | . . . 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 3989 | . . . 4 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
17 | cores 6102 | . . . 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 5107 | . 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹)) |
20 | 18 | oveq2i 7167 | . . . 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 4005 | . 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 1537 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 SubGrpcsubg 18273 DProd cdprd 19115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-gim 18399 df-cntz 18447 df-oppg 18474 df-cmn 18908 df-dprd 19117 |
This theorem is referenced by: (None) |
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