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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 2762 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 2 | resgrpplusfrn.o | . . . . 5 ⊢ 𝐹 = (+𝑓‘𝐻) | |
| 3 | 1, 2 | grpplusfo 19022 | . . . 4 ⊢ (𝐻 ∈ Grp → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
| 4 | 3 | adantr 485 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
| 5 | eqidd 2763 | . . . 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 17304 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻)) |
| 9 | 8 | adantl 486 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘𝐻)) |
| 10 | 9 | sqxpeqd 5692 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) = ((Base‘𝐻) × (Base‘𝐻))) |
| 11 | 5, 10, 9 | foeq123d 6813 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝐹:(𝑆 × 𝑆)–onto→𝑆 ↔ 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻))) |
| 12 | 4, 11 | mpbird 260 | . 2 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:(𝑆 × 𝑆)–onto→𝑆) |
| 13 | forn 6795 | . . 3 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → ran 𝐹 = 𝑆) | |
| 14 | 13 | eqcomd 2768 | . 2 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → 𝑆 = ran 𝐹) |
| 15 | 12, 14 | syl 18 | 1 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ⊆ wss 3904 × cxp 5658 ran crn 5661 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7412 Basecbs 17275 ↾s cress 17296 +𝑓cplusf 18701 Grpcgrp 19006 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-nn 12240 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-0g 17500 df-plusf 18703 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-grp 19009 |
| This theorem is used by: (None) |
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