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Theorem axtgsegcon 25163
 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 25152 . . . . . 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 3796 . . . . . . 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 3795 . . . . . . 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 3577 . . . . . 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 3598 . . . . 5 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3566 . . . 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 25155 . . . . . 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 479 . . . . 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 478 . . . 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 6556 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
1817eleq2d 2673 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
1918anbi1d 737 . . . . . . 7 (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
2019rexbidv 3034 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
21202ralbidv 2972 . . . . 5 (𝑥 = 𝑋 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
22 eleq1 2676 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23 oveq1 6556 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
2423eqeq1d 2612 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝑎 𝑏)))
2522, 24anbi12d 743 . . . . . . 7 (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2625rexbidv 3034 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
27262ralbidv 2972 . . . . 5 (𝑦 = 𝑌 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2821, 27rspc2v 3293 . . . 4 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) → ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2915, 16, 28syl2anc 691 . . 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 6556 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
3433eqeq2d 2620 . . . . . 6 (𝑎 = 𝐴 → ((𝑌 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝑏)))
3534anbi2d 736 . . . . 5 (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
3635rexbidv 3034 . . . 4 (𝑎 = 𝐴 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
37 oveq2 6557 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
3837eqeq2d 2620 . . . . . 6 (𝑏 = 𝐵 → ((𝑌 𝑧) = (𝐴 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝐵)))
3938anbi2d 736 . . . . 5 (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4039rexbidv 3034 . . . 4 (𝑏 = 𝐵 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4136, 40rspc2v 3293 . . 3 ((𝐴𝑃𝐵𝑃) → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4231, 32, 41syl2anc 691 . 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 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∩ cin 3539  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  TarskiGCcstrkgc 25130  TarskiGBcstrkgb 25131  TarskiGCBcstrkgcb 25132  Itvcitv 25135  LineGclng 25136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgcb 25149  df-trkg 25152 This theorem is referenced by:  tgcgrtriv  25179  tgbtwntriv2  25182  tgbtwnouttr2  25190  tgbtwndiff  25201  tgifscgr  25203  tgcgrxfr  25213  lnext  25262  tgbtwnconn1lem3  25269  tgbtwnconn1  25270  legtrid  25286  hlcgrex  25311  mirreu3  25349  miriso  25365  midexlem  25387  footex  25413  opphllem  25427  dfcgra2  25521  f1otrg  25551
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