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Theorem isnumbasgrplem2 37991
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 15924 . . 3 Base Fn V
2 ssv 3658 . . 3 Grp ⊆ V
3 fvelimab 6292 . . 3 ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
41, 2, 3mp2an 708 . 2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5 harcl 8507 . . . . . 6 (har‘𝑆) ∈ On
6 onenon 8813 . . . . . 6 ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
75, 6ax-mp 5 . . . . 5 (har‘𝑆) ∈ dom card
8 xpnum 8815 . . . . 5 (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
97, 7, 8mp2an 708 . . . 4 ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10 ssun1 3809 . . . . . . . 8 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11 simpr 476 . . . . . . . 8 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
1210, 11syl5sseqr 3687 . . . . . . 7 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13 fvex 6239 . . . . . . . 8 (Base‘𝑥) ∈ V
1413ssex 4835 . . . . . . 7 (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
1512, 14syl 17 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
167a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17 simp1l 1105 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18123ad2ant1 1102 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19 simp2 1082 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎𝑆)
2018, 19sseldd 3637 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21 ssun2 3810 . . . . . . . . . . 11 (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
2221, 11syl5sseqr 3687 . . . . . . . . . 10 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23223ad2ant1 1102 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24 simp3 1083 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
2523, 24sseldd 3637 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26 eqid 2651 . . . . . . . . 9 (Base‘𝑥) = (Base‘𝑥)
27 eqid 2651 . . . . . . . . 9 (+g𝑥) = (+g𝑥)
2826, 27grpcl 17477 . . . . . . . 8 ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
2917, 20, 25, 28syl3anc 1366 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
30 simp1r 1106 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
3129, 30eleqtrd 2732 . . . . . 6 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32 simplll 813 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
3322ad2antrr 762 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34 simprl 809 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
3533, 34sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36 simprr 811 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
3733, 36sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
3812ad2antrr 762 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39 simplr 807 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎𝑆)
4038, 39sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
4126, 27grplcan 17524 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
4232, 35, 37, 40, 41syl13anc 1368 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
43 simplll 813 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑥 ∈ Grp)
4412ad2antrr 762 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45 simprr 811 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑𝑆)
4644, 45sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑 ∈ (Base‘𝑥))
47 simprl 809 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎𝑆)
4844, 47sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎 ∈ (Base‘𝑥))
4922ad2antrr 762 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50 simplr 807 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (har‘𝑆))
5149, 50sseldd 3637 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (Base‘𝑥))
5226, 27grprcan 17502 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
5343, 46, 48, 51, 52syl13anc 1368 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
54 harndom 8510 . . . . . . 7 ¬ (har‘𝑆) ≼ 𝑆
5554a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
5615, 16, 16, 31, 42, 53, 55unxpwdom3 37982 . . . . 5 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆* ((har‘𝑆) × (har‘𝑆)))
57 wdomnumr 8925 . . . . . 6 (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
589, 57ax-mp 5 . . . . 5 (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
5956, 58sylib 208 . . . 4 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60 numdom 8899 . . . 4 ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
619, 59, 60sylancr 696 . . 3 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
6261rexlimiva 3057 . 2 (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
634, 62sylbi 207 1 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  cun 3605  wss 3607   class class class wbr 4685   × cxp 5141  dom cdm 5143  cima 5146  Oncon0 5761   Fn wfn 5921  cfv 5926  (class class class)co 6690  cdom 7995  harchar 8502  * cwdom 8503  cardccrd 8799  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-wdom 8505  df-card 8803  df-acn 8806  df-slot 15908  df-base 15910  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  isnumbasabl  37993  isnumbasgrp  37994
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