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Theorem axtgsegcon 28486
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 28475 . . . . . 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 4245 . . . . . . 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 4244 . . . . . . 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 4004 . . . . . 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 4029 . . . . 5 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3992 . . . 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 28478 . . . . . 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 496 . . . . 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 495 . . . 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 7437 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
1817eleq2d 2824 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
1918anbi1d 631 . . . . . . 7 (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
2019rexbidv 3176 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
21202ralbidv 3218 . . . . 5 (𝑥 = 𝑋 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
22 eleq1 2826 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23 oveq1 7437 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
2423eqeq1d 2736 . . . . . . . 8 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝑎 𝑏)))
2522, 24anbi12d 632 . . . . . . 7 (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2625rexbidv 3176 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
27262ralbidv 3218 . . . . 5 (𝑦 = 𝑌 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2821, 27rspc2v 3632 . . . 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 7437 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
3433eqeq2d 2745 . . . . . 6 (𝑎 = 𝐴 → ((𝑌 𝑧) = (𝑎 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝑏)))
3534anbi2d 630 . . . . 5 (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
3635rexbidv 3176 . . . 4 (𝑎 = 𝐴 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
37 oveq2 7438 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
3837eqeq2d 2745 . . . . . 6 (𝑏 = 𝐵 → ((𝑌 𝑧) = (𝐴 𝑏) ↔ (𝑌 𝑧) = (𝐴 𝐵)))
3938anbi2d 630 . . . . 5 (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4039rexbidv 3176 . . . 4 (𝑏 = 𝐵 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4136, 40rspc2v 3632 . . 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 395  w3o 1085  w3a 1086   = wceq 1536  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  [wsbc 3790  cdif 3959  cin 3961  {csn 4630  cfv 6562  (class class class)co 7430  cmpo 7432  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  TarskiGCcstrkgc 28450  TarskiGBcstrkgb 28451  TarskiGCBcstrkgcb 28452  Itvcitv 28455  LineGclng 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgcb 28472  df-trkg 28475
This theorem is referenced by:  tgcgrtriv  28506  tgbtwntriv2  28509  tgbtwnouttr2  28517  tgbtwndiff  28528  tgifscgr  28530  tgcgrxfr  28540  lnext  28589  tgbtwnconn1lem3  28596  tgbtwnconn1  28597  legtrid  28613  hlcgrex  28638  mirreu3  28676  miriso  28692  midexlem  28714  footexALT  28740  footex  28743  opphllem  28757  flatcgra  28846  dfcgra2  28852  f1otrg  28893
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