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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 8410 | . . 3 ⊢ (𝐶 ≈ 𝐵 ↔ 𝐵 ≈ 𝐶) | |
2 | bren 8371 | . . 3 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) | |
3 | 1, 2 | bitri 276 | . 2 ⊢ (𝐶 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) |
4 | eqidd 2796 | . . . . . . 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 6495 | . . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝑓:𝐵–onto→𝐶) | |
8 | 7 | adantr 481 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓:𝐵–onto→𝐶) |
9 | simpr 485 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑅 ∈ Abel) | |
10 | 4, 6, 8, 9 | imasbas 16619 | . . . . . 6 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝐶 = (Base‘(𝑓 “s 𝑅))) |
11 | simpl 483 | . . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓:𝐵–1-1-onto→𝐶) | |
12 | ablgrp 18643 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) | |
13 | 12 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑅 ∈ Grp) |
14 | 4, 6, 11, 13 | imasgim 39211 | . . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝑓 ∈ (𝑅 GrpIso (𝑓 “s 𝑅))) |
15 | brgici 18156 | . . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 GrpIso (𝑓 “s 𝑅)) → 𝑅 ≃𝑔 (𝑓 “s 𝑅)) | |
16 | gicabl 39210 | . . . . . . . . 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 233 | . . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → (𝑓 “s 𝑅) ∈ Abel) |
19 | basfn 16337 | . . . . . . . 8 ⊢ Base Fn V | |
20 | ssv 3916 | . . . . . . . 8 ⊢ Abel ⊆ V | |
21 | fnfvima 6865 | . . . . . . . 8 ⊢ ((Base Fn V ∧ Abel ⊆ V ∧ (𝑓 “s 𝑅) ∈ Abel) → (Base‘(𝑓 “s 𝑅)) ∈ (Base “ Abel)) | |
22 | 19, 20, 21 | mp3an12 1443 | . . . . . . 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 2883 | . . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑅 ∈ Abel) → 𝐶 ∈ (Base “ Abel)) |
25 | 24 | ex 413 | . . . 4 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → (𝑅 ∈ Abel → 𝐶 ∈ (Base “ Abel))) |
26 | 25 | exlimiv 1908 | . . 3 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐶 → (𝑅 ∈ Abel → 𝐶 ∈ (Base “ Abel))) |
27 | 26 | impcom 408 | . 2 ⊢ ((𝑅 ∈ Abel ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐶 ∈ (Base “ Abel)) |
28 | 3, 27 | sylan2b 593 | 1 ⊢ ((𝑅 ∈ Abel ∧ 𝐶 ≈ 𝐵) → 𝐶 ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3863 class class class wbr 4966 “ cima 5451 Fn wfn 6225 –onto→wfo 6228 –1-1-onto→wf1o 6229 ‘cfv 6230 (class class class)co 7021 ≈ cen 8359 Basecbs 16317 “s cimas 16611 Grpcgrp 17866 GrpIso cgim 18143 ≃𝑔 cgic 18144 Abelcabl 18639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-inf 8758 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-0g 16549 df-imas 16615 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-grp 17869 df-minusg 17870 df-ghm 18102 df-gim 18145 df-gic 18146 df-cmn 18640 df-abl 18641 |
This theorem is referenced by: isnumbasgrplem3 39216 |
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