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Theorem axtgsegcon 26177
Description: Axiom of segment construction, Axiom A4 of [Schwabhauser] p. 11. As discussed in Axiom 4 of [Tarski1999] p. 178, "The intuitive content [is that] given any line segment 𝐴𝐵, one can construct a line segment congruent to it, starting at any point 𝑌 and going in the direction of any ray containing 𝑌. The ray is determined by the point 𝑌 and a second point 𝑋, the endpoint of the ray. The other endpoint of the line segment to be constructed is just the point 𝑧 whose existence is asserted." (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Base‘𝐺)
axtrkg.d = (dist‘𝐺)
axtrkg.i 𝐼 = (Itv‘𝐺)
axtrkg.g (𝜑𝐺 ∈ TarskiG)
axtgsegcon.1 (𝜑𝑋𝑃)
axtgsegcon.2 (𝜑𝑌𝑃)
axtgsegcon.3 (𝜑𝐴𝑃)
axtgsegcon.4 (𝜑𝐵𝑃)
Assertion
Ref Expression
axtgsegcon (𝜑 → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐼   𝑧,𝑃   𝑧,𝑋   𝑧,𝑌   𝑧,
Allowed substitution hints:   𝜑(𝑧)   𝐺(𝑧)

Proof of Theorem axtgsegcon
Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑎 𝑏 𝑐 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 26166 . . . . . 6 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2 inss2 4203 . . . . . . 7 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
3 inss1 4202 . . . . . . 7 (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}) ⊆ TarskiGCB
42, 3sstri 3973 . . . . . 6 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGCB
51, 4eqsstri 3998 . . . . 5 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3962 . . . 4 (𝜑𝐺 ∈ TarskiGCB)
8 axtrkg.p . . . . . . 7 𝑃 = (Base‘𝐺)
9 axtrkg.d . . . . . . 7 = (dist‘𝐺)
10 axtrkg.i . . . . . . 7 𝐼 = (Itv‘𝐺)
118, 9, 10istrkgcb 26169 . . . . . 6 (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))))
1211simprbi 497 . . . . 5 (𝐺 ∈ TarskiGCB → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
1312simprd 496 . . . 4 (𝐺 ∈ TarskiGCB → ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))
147, 13syl 17 . . 3 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))
15 axtgsegcon.1 . . . 4 (𝜑𝑋𝑃)
16 axtgsegcon.2 . . . 4 (𝜑𝑌𝑃)
17 oveq1 7152 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
1817eleq2d 2895 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
1918anbi1d 629 . . . . . . 7 (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
2019rexbidv 3294 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
21202ralbidv 3196 . . . . 5 (𝑥 = 𝑋 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
22 eleq1 2897 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23 oveq1 7152 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
2423eqeq1d 2820 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝑎 𝑏)))
2522, 24anbi12d 630 . . . . . . 7 (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2625rexbidv 3294 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
27262ralbidv 3196 . . . . 5 (𝑦 = 𝑌 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2821, 27rspc2v 3630 . . . 4 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) → ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2915, 16, 28syl2anc 584 . . 3 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) → ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
3014, 29mpd 15 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)))
31 axtgsegcon.3 . . 3 (𝜑𝐴𝑃)
32 axtgsegcon.4 . . 3 (𝜑𝐵𝑃)
33 oveq1 7152 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
3433eqeq2d 2829 . . . . . 6 (𝑎 = 𝐴 → ((𝑌 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝑏)))
3534anbi2d 628 . . . . 5 (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
3635rexbidv 3294 . . . 4 (𝑎 = 𝐴 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
37 oveq2 7153 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
3837eqeq2d 2829 . . . . . 6 (𝑏 = 𝐵 → ((𝑌 𝑧) = (𝐴 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝐵)))
3938anbi2d 628 . . . . 5 (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4039rexbidv 3294 . . . 4 (𝑏 = 𝐵 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4136, 40rspc2v 3630 . . 3 ((𝐴𝑃𝐵𝑃) → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4231, 32, 41syl2anc 584 . 2 (𝜑 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4330, 42mpd 15 1 (𝜑 → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  [wsbc 3769  cdif 3930  cin 3932  {csn 4557  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  TarskiGCcstrkgc 26144  TarskiGBcstrkgb 26145  TarskiGCBcstrkgcb 26146  Itvcitv 26149  LineGclng 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  tgcgrtriv  26197  tgbtwntriv2  26200  tgbtwnouttr2  26208  tgbtwndiff  26219  tgifscgr  26221  tgcgrxfr  26231  lnext  26280  tgbtwnconn1lem3  26287  tgbtwnconn1  26288  legtrid  26304  hlcgrex  26329  mirreu3  26367  miriso  26383  midexlem  26405  footexALT  26431  footex  26434  opphllem  26448  flatcgra  26537  dfcgra2  26543  f1otrg  26584
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