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Theorem axtgsegcon 26233
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 26222 . . . . . 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 4180 . . . . . . 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 4179 . . . . . . 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 3951 . . . . . 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 3976 . . . . 5 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3940 . . . 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 26225 . . . . . 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 499 . . . . 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 498 . . . 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 7136 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
1817eleq2d 2896 . . . . . . . 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 3282 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
21202ralbidv 3186 . . . . 5 (𝑥 = 𝑋 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
22 eleq1 2898 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23 oveq1 7136 . . . . . . . . 9 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
2423eqeq1d 2822 . . . . . . . 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 3282 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
27262ralbidv 3186 . . . . 5 (𝑦 = 𝑌 → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) ↔ ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2821, 27rspc2v 3609 . . . 4 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)) → ∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏))))
2915, 16, 28syl2anc 586 . . 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 7136 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
3433eqeq2d 2831 . . . . . 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 3282 . . . 4 (𝑎 = 𝐴 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏))))
37 oveq2 7137 . . . . . . 7 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
3837eqeq2d 2831 . . . . . 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 3282 . . . 4 (𝑏 = 𝐵 → (∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝑏)) ↔ ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4136, 40rspc2v 3609 . . 3 ((𝐴𝑃𝐵𝑃) → (∀𝑎𝑃𝑏𝑃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝑎 𝑏)) → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵))))
4231, 32, 41syl2anc 586 . 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 398  w3o 1082  w3a 1083   = wceq 1537  wcel 2114  {cab 2798  wne 3006  wral 3125  wrex 3126  {crab 3129  Vcvv 3470  [wsbc 3748  cdif 3906  cin 3908  {csn 4539  cfv 6327  (class class class)co 7129  cmpo 7131  Basecbs 16458  distcds 16549  TarskiGcstrkg 26199  TarskiGCcstrkgc 26200  TarskiGBcstrkgb 26201  TarskiGCBcstrkgcb 26202  Itvcitv 26205  LineGclng 26206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-nul 5182
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-iota 6286  df-fv 6335  df-ov 7132  df-trkgcb 26219  df-trkg 26222
This theorem is referenced by:  tgcgrtriv  26253  tgbtwntriv2  26256  tgbtwnouttr2  26264  tgbtwndiff  26275  tgifscgr  26277  tgcgrxfr  26287  lnext  26336  tgbtwnconn1lem3  26343  tgbtwnconn1  26344  legtrid  26360  hlcgrex  26385  mirreu3  26423  miriso  26439  midexlem  26461  footexALT  26487  footex  26490  opphllem  26504  flatcgra  26593  dfcgra2  26599  f1otrg  26640
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