Theorem axtgsegcon 28490

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgsegcon.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgsegcon.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
axtgsegcon.3	⊢ (𝜑 → 𝐴 ∈ 𝑃)
axtgsegcon.4	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
axtgsegcon	⊢ (𝜑 → ∃𝑧 ∈ 𝑃 (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.

Step	Hyp	Ref	Expression
1		df-trkg 28479	. . . . . 6 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2		inss2 4259	. . . . . . 7 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
3		inss1 4258	. . . . . . 7 ⊢ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}) ⊆ TarskiGCB
4	2, 3	sstri 4018	. . . . . 6 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGCB
5	1, 4	eqsstri 4043	. . . . 5 ⊢ TarskiG ⊆ TarskiGCB
6		axtrkg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 4006	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiGCB)
8		axtrkg.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
9		axtrkg.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
10		axtrkg.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
11	8, 9, 10	istrkgcb 28482	. . . . . 6 ⊢ (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∀𝑐 ∈ 𝑃 ∀𝑣 ∈ 𝑃 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 − 𝑦) = (𝑎 − 𝑏) ∧ (𝑦 − 𝑧) = (𝑏 − 𝑐)) ∧ ((𝑥 − 𝑢) = (𝑎 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑏 − 𝑣)))) → (𝑧 − 𝑢) = (𝑐 − 𝑣)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)))))
12	11	simprbi 496	. . . . 5 ⊢ (𝐺 ∈ TarskiGCB → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∀𝑐 ∈ 𝑃 ∀𝑣 ∈ 𝑃 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 − 𝑦) = (𝑎 − 𝑏) ∧ (𝑦 − 𝑧) = (𝑏 − 𝑐)) ∧ ((𝑥 − 𝑢) = (𝑎 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑏 − 𝑣)))) → (𝑧 − 𝑢) = (𝑐 − 𝑣)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
13	12	simprd 495	. . . 4 ⊢ (𝐺 ∈ TarskiGCB → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)))
14	7, 13	syl 17	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)))
15		axtgsegcon.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
16		axtgsegcon.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
17		oveq1 7455	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
18	17	eleq2d 2830	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
19	18	anbi1d 630	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
20	19	rexbidv 3185	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
21	20	2ralbidv 3227	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
22		eleq1 2832	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23		oveq1 7455	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑧) = (𝑌 − 𝑧))
24	23	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑦 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝑎 − 𝑏)))
25	22, 24	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
26	25	rexbidv 3185	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
27	26	2ralbidv 3227	. . . . 5 ⊢ (𝑦 = 𝑌 → (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
28	21, 27	rspc2v 3646	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
29	15, 16, 28	syl2anc 583	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
30	14, 29	mpd 15	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)))
31		axtgsegcon.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
32		axtgsegcon.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
33		oveq1 7455	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏))
34	33	eqeq2d 2751	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑌 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝑏)))
35	34	anbi2d 629	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
36	35	rexbidv 3185	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
37		oveq2 7456	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵))
38	37	eqeq2d 2751	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑌 − 𝑧) = (𝐴 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝐵)))
39	38	anbi2d 629	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
40	39	rexbidv 3185	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
41	36, 40	rspc2v 3646	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) → ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
42	31, 32, 41	syl2anc 583	. 2 ⊢ (𝜑 → (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) → ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
43	30, 42	mpd 15	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488  [wsbc 3804   ∖ cdif 3973   ∩ cin 3975  {csn 4648  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  TarskiG_Ccstrkgc 28454  TarskiG_Bcstrkgb 28455  TarskiG_CBcstrkgcb 28456  Itvcitv 28459  LineGclng 28460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-trkgcb 28476  df-trkg 28479

This theorem is referenced by:  tgcgrtriv  28510  tgbtwntriv2  28513  tgbtwnouttr2  28521  tgbtwndiff  28532  tgifscgr  28534  tgcgrxfr  28544  lnext  28593  tgbtwnconn1lem3  28600  tgbtwnconn1  28601  legtrid  28617  hlcgrex  28642  mirreu3  28680  miriso  28696  midexlem  28718  footexALT  28744  footex  28747  opphllem  28761  flatcgra  28850  dfcgra2  28856  f1otrg  28897