![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resgrpplusfrn | Structured version Visualization version GIF version |
Description: The underlying set of a group operation which is a restriction of a structure. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by AV, 30-Aug-2021.) |
Ref | Expression |
---|---|
resgrpplusfrn.b | ⊢ 𝐵 = (Base‘𝐺) |
resgrpplusfrn.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
resgrpplusfrn.o | ⊢ 𝐹 = (+𝑓‘𝐻) |
Ref | Expression |
---|---|
resgrpplusfrn | ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
2 | resgrpplusfrn.o | . . . . 5 ⊢ 𝐹 = (+𝑓‘𝐻) | |
3 | 1, 2 | grpplusfo 18764 | . . . 4 ⊢ (𝐻 ∈ Grp → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
5 | eqidd 2738 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹 = 𝐹) | |
6 | resgrpplusfrn.h | . . . . . . 7 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
7 | resgrpplusfrn.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
8 | 6, 7 | ressbas2 17121 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻)) |
9 | 8 | adantl 483 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘𝐻)) |
10 | 9 | sqxpeqd 5666 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) = ((Base‘𝐻) × (Base‘𝐻))) |
11 | 5, 10, 9 | foeq123d 6778 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝐹:(𝑆 × 𝑆)–onto→𝑆 ↔ 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻))) |
12 | 4, 11 | mpbird 257 | . 2 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:(𝑆 × 𝑆)–onto→𝑆) |
13 | forn 6760 | . . 3 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → ran 𝐹 = 𝑆) | |
14 | 13 | eqcomd 2743 | . 2 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → 𝑆 = ran 𝐹) |
15 | 12, 14 | syl 17 | 1 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 × cxp 5632 ran crn 5635 –onto→wfo 6495 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 ↾s cress 17113 +𝑓cplusf 18495 Grpcgrp 18749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-1cn 11110 ax-addcl 11112 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12155 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-0g 17324 df-plusf 18497 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |