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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 2727 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
2 | resgrpplusfrn.o | . . . . 5 ⊢ 𝐹 = (+𝑓‘𝐻) | |
3 | 1, 2 | grpplusfo 18897 | . . . 4 ⊢ (𝐻 ∈ Grp → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐹:((Base‘𝐻) × (Base‘𝐻))–onto→(Base‘𝐻)) |
5 | eqidd 2728 | . . . 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 17209 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻)) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘𝐻)) |
10 | 9 | sqxpeqd 5704 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) = ((Base‘𝐻) × (Base‘𝐻))) |
11 | 5, 10, 9 | foeq123d 6826 | . . 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 6808 | . . 3 ⊢ (𝐹:(𝑆 × 𝑆)–onto→𝑆 → ran 𝐹 = 𝑆) | |
14 | 13 | eqcomd 2733 | . 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 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 × cxp 5670 ran crn 5673 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 ↾s cress 17200 +𝑓cplusf 18588 Grpcgrp 18881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-1cn 11188 ax-addcl 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12235 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-0g 17414 df-plusf 18590 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 |
This theorem is referenced by: (None) |
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