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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 2821 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
2 | resgrpplusfrn.o | . . . . 5 ⊢ 𝐹 = (+𝑓‘𝐻) | |
3 | 1, 2 | grpplusfo 18110 | . . . 4 ⊢ (𝐻 ∈ Grp → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
5 | eqidd 2822 | . . . 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 16549 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻)) |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘𝐻)) |
10 | 9 | sqxpeqd 5581 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) = ((Base‘𝐻) × (Base‘𝐻))) |
11 | 5, 10, 9 | foeq123d 6603 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝐹:(𝑆 × 𝑆)–onto→𝑆 ↔ 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻))) |
12 | 4, 11 | mpbird 259 | . 2 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:(𝑆 × 𝑆)–onto→𝑆) |
13 | forn 6587 | . . 3 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → ran 𝐹 = 𝑆) | |
14 | 13 | eqcomd 2827 | . 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 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 × cxp 5547 ran crn 5550 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 +𝑓cplusf 17843 Grpcgrp 18097 |
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-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
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-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-iun 4913 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-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-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-0g 16709 df-plusf 17845 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 |
This theorem is referenced by: (None) |
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