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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasgrplem1 | Structured version Visualization version GIF version |
Description: A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasgrplem1.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
isnumbasgrplem1 | ⊢ ((𝑅 ∈ Abel ∧ 𝐶 ≈ 𝐵) → 𝐶 ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymb 8949 | . . 3 ⊢ (𝐶 ≈ 𝐵 ↔ 𝐵 ≈ 𝐶) | |
2 | bren 8900 | . . 3 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (𝐶 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) |
4 | eqidd 2738 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → (𝑓 “s 𝑅) = (𝑓 “s 𝑅)) | |
5 | isnumbasgrplem1.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝐵 = (Base‘𝑅)) |
7 | f1ofo 6796 | . . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝑓:𝐵–onto→𝐶) | |
8 | 7 | adantr 482 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓:𝐵–onto→𝐶) |
9 | simpr 486 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑅 ∈ Abel) | |
10 | 4, 6, 8, 9 | imasbas 17401 | . . . . . 6 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝐶 = (Base‘(𝑓 “s 𝑅))) |
11 | simpl 484 | . . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓:𝐵–1-1-onto→𝐶) | |
12 | ablgrp 19574 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) | |
13 | 12 | adantl 483 | . . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑅 ∈ Grp) |
14 | 4, 6, 11, 13 | imasgim 41456 | . . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓 ∈ (𝑅 GrpIso (𝑓 “s 𝑅))) |
15 | brgici 19067 | . . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 GrpIso (𝑓 “s 𝑅)) → 𝑅 ≃𝑔 (𝑓 “s 𝑅)) | |
16 | gicabl 41455 | . . . . . . . . 9 ⊢ (𝑅 ≃𝑔 (𝑓 “s 𝑅) → (𝑅 ∈ Abel ↔ (𝑓 “s 𝑅) ∈ Abel)) | |
17 | 14, 15, 16 | 3syl 18 | . . . . . . . 8 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → (𝑅 ∈ Abel ↔ (𝑓 “s 𝑅) ∈ Abel)) |
18 | 9, 17 | mpbid 231 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → (𝑓 “s 𝑅) ∈ Abel) |
19 | basfn 17094 | . . . . . . . 8 ⊢ Base Fn V | |
20 | ssv 3973 | . . . . . . . 8 ⊢ Abel ⊆ V | |
21 | fnfvima 7188 | . . . . . . . 8 ⊢ ((Base Fn V ∧ Abel ⊆ V ∧ (𝑓 “s 𝑅) ∈ Abel) → (Base‘(𝑓 “s 𝑅)) ∈ (Base “ Abel)) | |
22 | 19, 20, 21 | mp3an12 1452 | . . . . . . 7 ⊢ ((𝑓 “s 𝑅) ∈ Abel → (Base‘(𝑓 “s 𝑅)) ∈ (Base “ Abel)) |
23 | 18, 22 | syl 17 | . . . . . 6 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → (Base‘(𝑓 “s 𝑅)) ∈ (Base “ Abel)) |
24 | 10, 23 | eqeltrd 2838 | . . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝐶 ∈ (Base “ Abel)) |
25 | 24 | ex 414 | . . . 4 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → (𝑅 ∈ Abel → 𝐶 ∈ (Base “ Abel))) |
26 | 25 | exlimiv 1934 | . . 3 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐶 → (𝑅 ∈ Abel → 𝐶 ∈ (Base “ Abel))) |
27 | 26 | impcom 409 | . 2 ⊢ ((𝑅 ∈ Abel ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐶 ∈ (Base “ Abel)) |
28 | 3, 27 | sylan2b 595 | 1 ⊢ ((𝑅 ∈ Abel ∧ 𝐶 ≈ 𝐵) → 𝐶 ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 class class class wbr 5110 “ cima 5641 Fn wfn 6496 –onto→wfo 6499 –1-1-onto→wf1o 6500 ‘cfv 6501 (class class class)co 7362 ≈ cen 8887 Basecbs 17090 “s cimas 17393 Grpcgrp 18755 GrpIso cgim 19054 ≃𝑔 cgic 19055 Abelcabl 19570 |
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-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-0g 17330 df-imas 17397 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-ghm 19013 df-gim 19056 df-gic 19057 df-cmn 19571 df-abl 19572 |
This theorem is referenced by: isnumbasgrplem3 41461 |
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