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Theorem axtgcont 23058
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p  |-  P  =  ( Base `  G
)
axtrkg.d  |-  .-  =  ( dist `  G )
axtrkg.i  |-  I  =  (Itv `  G )
axtrkg.g  |-  ( ph  ->  G  e. TarskiG )
axtgcont.1  |-  ( ph  ->  S  C_  P )
axtgcont.2  |-  ( ph  ->  T  C_  P )
axtgcont.3  |-  ( ph  ->  A  e.  P )
axtgcont.4  |-  ( (
ph  /\  u  e.  S  /\  v  e.  T
)  ->  u  e.  ( A I v ) )
Assertion
Ref Expression
axtgcont  |-  ( ph  ->  E. b  e.  P  A. x  e.  S  A. y  e.  T  b  e.  ( x I y ) )
Distinct variable groups:    x, y    v, b, A, u, x, y    I, b    v, u, x, y, I    P, b, u, v, x, y    S, b, x    T, b, x, y    .- , b, u, v, x, y    ph, u, v    u, S, v    u, T, v    u, A, x, y
Allowed substitution hints:    ph( x, y, b)    S( y)    G( x, y, v, u, b)

Proof of Theorem axtgcont
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 axtgcont.3 . . 3  |-  ( ph  ->  A  e.  P )
2 axtgcont.4 . . . . 5  |-  ( (
ph  /\  u  e.  S  /\  v  e.  T
)  ->  u  e.  ( A I v ) )
323expb 1189 . . . 4  |-  ( (
ph  /\  ( u  e.  S  /\  v  e.  T ) )  ->  u  e.  ( A I v ) )
43ralrimivva 2908 . . 3  |-  ( ph  ->  A. u  e.  S  A. v  e.  T  u  e.  ( A I v ) )
5 simplr 754 . . . . . . 7  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  x  =  u )
6 simpll 753 . . . . . . . 8  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  a  =  A )
7 simpr 461 . . . . . . . 8  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  y  =  v )
86, 7oveq12d 6213 . . . . . . 7  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  (
a I y )  =  ( A I v ) )
95, 8eleq12d 2534 . . . . . 6  |-  ( ( ( a  =  A  /\  x  =  u )  /\  y  =  v )  ->  (
x  e.  ( a I y )  <->  u  e.  ( A I v ) ) )
109cbvraldva 3053 . . . . 5  |-  ( ( a  =  A  /\  x  =  u )  ->  ( A. y  e.  T  x  e.  ( a I y )  <->  A. v  e.  T  u  e.  ( A I v ) ) )
1110cbvraldva 3053 . . . 4  |-  ( a  =  A  ->  ( A. x  e.  S  A. y  e.  T  x  e.  ( a
I y )  <->  A. u  e.  S  A. v  e.  T  u  e.  ( A I v ) ) )
1211rspcev 3173 . . 3  |-  ( ( A  e.  P  /\  A. u  e.  S  A. v  e.  T  u  e.  ( A I v ) )  ->  E. a  e.  P  A. x  e.  S  A. y  e.  T  x  e.  ( a I y ) )
131, 4, 12syl2anc 661 . 2  |-  ( ph  ->  E. a  e.  P  A. x  e.  S  A. y  e.  T  x  e.  ( a
I y ) )
14 axtrkg.p . . 3  |-  P  =  ( Base `  G
)
15 axtrkg.d . . 3  |-  .-  =  ( dist `  G )
16 axtrkg.i . . 3  |-  I  =  (Itv `  G )
17 axtrkg.g . . 3  |-  ( ph  ->  G  e. TarskiG )
18 axtgcont.1 . . 3  |-  ( ph  ->  S  C_  P )
19 axtgcont.2 . . 3  |-  ( ph  ->  T  C_  P )
2014, 15, 16, 17, 18, 19axtgcont1 23057 . 2  |-  ( ph  ->  ( E. a  e.  P  A. x  e.  S  A. y  e.  T  x  e.  ( a I y )  ->  E. b  e.  P  A. x  e.  S  A. y  e.  T  b  e.  ( x I y ) ) )
2113, 20mpd 15 1  |-  ( ph  ->  E. b  e.  P  A. x  e.  S  A. y  e.  T  b  e.  ( x I y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797    C_ wss 3431   ` cfv 5521  (class class class)co 6195   Basecbs 14287   distcds 14361  TarskiGcstrkg 23017  Itvcitv 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198  df-trkgb 23037  df-trkg 23042
This theorem is referenced by:  f1otrg  23264
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