Theorem fingrphf 46229

Description: Every finite group is isomorphic to a group with hereditarily finite components. (Contributed by Eric Schmidt, 24-Oct-2026.)

Hypotheses

Ref	Expression
fingrphf.1	⊢ 𝐵 = (Base‘𝐺)
fingrphf.2	⊢ (𝜑 → 𝐺 ∈ Grp)
fingrphf.3	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
fingrphf	⊢ (𝜑 → ∃ℎ ∈ HFStruct 𝐺 ≃𝑔 ℎ)

Distinct variable groups:   𝐵,ℎ   ℎ,𝐺

Allowed substitution hint:   𝜑(ℎ)

Proof of Theorem fingrphf

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fingrphf.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		ficardid 10067	. . . . 5 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
3	2	ensymd 9057	. . . 4 ⊢ (𝐵 ∈ Fin → 𝐵 ≈ (card‘𝐵))
4		bren 9008	. . . 4 ⊢ (𝐵 ≈ (card‘𝐵) ↔ ∃𝑓 𝑓:𝐵⟶1-1onto(card‘𝐵))
5	3, 4	sylib 221	. . 3 ⊢ (𝐵 ∈ Fin → ∃𝑓 𝑓:𝐵⟶1-1onto(card‘𝐵))
6	1, 5	syl 18	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:𝐵⟶1-1onto(card‘𝐵))
7		breq2 5110	. . 3 ⊢ (ℎ = {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩} → (𝐺 ≃𝑔 ℎ ↔ 𝐺 ≃𝑔 {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}))
8		eqid 2763	. . . 4 ⊢ (Base‘(𝑓 “s 𝐺)) = (Base‘(𝑓 “s 𝐺))
9		eqid 2763	. . . 4 ⊢ (+𝑓‘(𝑓 “s 𝐺)) = (+𝑓‘(𝑓 “s 𝐺))
10		eqid 2763	. . . 4 ⊢ {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩} = {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}
11		fingrphf.2	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
12		eqidd 2764	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (𝑓 “s 𝐺) = (𝑓 “s 𝐺))
13		fingrphf.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
14	13	a1i 11	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐵 = (Base‘𝐺))
15		simpr 490	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝑓:𝐵⟶1-1onto(card‘𝐵))
16		simpl 488	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐺 ∈ Grp)
17		fvex 6918	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
18	12, 14, 15, 16, 17	imasgimw 46120	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝑓 ∈ (𝐺 GrpIso (𝑓 “s 𝐺)))
19		gimghm 19504	. . . . . 6 ⊢ (𝑓 ∈ (𝐺 GrpIso (𝑓 “s 𝐺)) → 𝑓 ∈ (𝐺 GrpHom (𝑓 “s 𝐺)))
20		ghmgrp2 19459	. . . . . 6 ⊢ (𝑓 ∈ (𝐺 GrpHom (𝑓 “s 𝐺)) → (𝑓 “s 𝐺) ∈ Grp)
21	18, 19, 20	3syl 19	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (𝑓 “s 𝐺) ∈ Grp)
22	11, 21	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (𝑓 “s 𝐺) ∈ Grp)
23		eqidd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (𝑓 “s 𝐺) = (𝑓 “s 𝐺))
24	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐵 = (Base‘𝐺))
25		f1ofo 6851	. . . . . . 7 ⊢ (𝑓:𝐵⟶1-1onto(card‘𝐵) → 𝑓:𝐵⟶onto(card‘𝐵))
26	25	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝑓:𝐵⟶onto(card‘𝐵))
27	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐺 ∈ Grp)
28	23, 24, 26, 27	imasbas 17710	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (card‘𝐵) = (Base‘(𝑓 “s 𝐺)))
29		ficardom 10066	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
30		omhf 46214	. . . . . . 7 ⊢ ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ HF )
31	1, 29, 30	3syl 19	. . . . . 6 ⊢ (𝜑 → (card‘𝐵) ∈ HF )
32	31	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (card‘𝐵) ∈ HF )
33	28, 32	eqeltrrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (Base‘(𝑓 “s 𝐺)) ∈ HF )
34	8, 9, 10, 22, 33	grpcanonhfstruct 46228	. . 3 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩} ∈ HFStruct)
35	8, 9, 10	grpgimcanon 46121	. . . . . . 7 ⊢ ((𝑓 “s 𝐺) ∈ Grp → ( I ↾ (Base‘(𝑓 “s 𝐺))) ∈ ((𝑓 “s 𝐺) GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}))
36	21, 35	syl 18	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → ( I ↾ (Base‘(𝑓 “s 𝐺))) ∈ ((𝑓 “s 𝐺) GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}))
37		gimco 19508	. . . . . 6 ⊢ ((( I ↾ (Base‘(𝑓 “s 𝐺))) ∈ ((𝑓 “s 𝐺) GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}) ∧ 𝑓 ∈ (𝐺 GrpIso (𝑓 “s 𝐺))) → (( I ↾ (Base‘(𝑓 “s 𝐺))) ∘ 𝑓) ∈ (𝐺 GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}))
38	36, 18, 37	syl2anc 596	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → (( I ↾ (Base‘(𝑓 “s 𝐺))) ∘ 𝑓) ∈ (𝐺 GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}))
39		brgici 19512	. . . . 5 ⊢ ((( I ↾ (Base‘(𝑓 “s 𝐺))) ∘ 𝑓) ∈ (𝐺 GrpIso {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩}) → 𝐺 ≃𝑔 {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩})
40	38, 39	syl 18	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐺 ≃𝑔 {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩})
41	11, 40	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → 𝐺 ≃𝑔 {⟨(Base‘ndx), (Base‘(𝑓 “s 𝐺))⟩, ⟨(+g‘ndx), (+𝑓‘(𝑓 “s 𝐺))⟩})
42	7, 34, 41	rspcedvdw 3583	. 2 ⊢ ((𝜑 ∧ 𝑓:𝐵⟶1-1onto(card‘𝐵)) → ∃ℎ ∈ HFStruct 𝐺 ≃𝑔 ℎ)
43	6, 42	exlimddv 1970	1 ⊢ (𝜑 → ∃ℎ ∈ HFStruct 𝐺 ≃𝑔 ℎ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  {cpr 4590  ⟨cop 4838   class class class wbr 5106   I cid 5549   ↾ cres 5657   ∘ ccom 5659  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  ωcom 7898   ≈ cen 8995  Fincfn 8998   HF chf 9928  cardccrd 10040  ndxcnx 17397  Basecbs 17413  +_gcplusg 17454   “_s cimas 17702  +_𝑓cplusf 18839  Grpcgrp 19170   GrpHom cghm 19453   GrpIso cgim 19497   ≃_𝑔 cgic 19498  HFStructchfstruct 46219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-r1 9792  df-rank 9793  df-hf 9929  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-slot 17386  df-ndx 17398  df-base 17414  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-0g 17638  df-imas 17706  df-plusf 18841  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-grp 19173  df-minusg 19174  df-ghm 19454  df-gim 19499  df-gic 19500  df-hfstruct 46220

This theorem is used by: (None)