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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 18116 | . . . 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 16555 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻)) |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘𝐻)) |
10 | 9 | sqxpeqd 5587 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) = ((Base‘𝐻) × (Base‘𝐻))) |
11 | 5, 10, 9 | foeq123d 6609 | . . 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 6593 | . . 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 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 ran crn 5556 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 +𝑓cplusf 17849 Grpcgrp 18103 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
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-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-iun 4921 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-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-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-0g 16715 df-plusf 17851 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 |
This theorem is referenced by: (None) |
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