Theorem axtgsegcon 28709

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 28698	. . . . . 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 4189	. . . . . . 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 4188	. . . . . . 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 3945	. . . . . 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 3982	. . . . 5 ⊢ TarskiG ⊆ TarskiGCB
6		axtrkg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3934	. . . 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 28701	. . . . . 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 502	. . . . 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 500	. . . 4 ⊢ (𝐺 ∈ TarskiGCB → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)))
14	7, 13	syl 18	. . 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 7417	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
18	17	eleq2d 2847	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
19	18	anbi1d 642	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
20	19	rexbidv 3187	. . . . . 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 2849	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23		oveq1 7417	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑧) = (𝑌 − 𝑧))
24	23	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑦 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝑎 − 𝑏)))
25	22, 24	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
26	25	rexbidv 3187	. . . . . 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 3591	. . . 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 595	. . 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 16	. 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 7417	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏))
34	33	eqeq2d 2772	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑌 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝑏)))
35	34	anbi2d 641	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
36	35	rexbidv 3187	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
37		oveq2 7418	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵))
38	37	eqeq2d 2772	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑌 − 𝑧) = (𝐴 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝐵)))
39	38	anbi2d 641	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
40	39	rexbidv 3187	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
41	36, 40	rspc2v 3591	. . 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 595	. 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 16	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3414  Vcvv 3453  [wsbc 3743   ∖ cdif 3901   ∩ cin 3903  {csn 4588  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17268  distcds 17318  TarskiGcstrkg 28672  TarskiG_Ccstrkgc 28673  TarskiG_Bcstrkgb 28674  TarskiG_CBcstrkgcb 28675  Itvcitv 28678  LineGclng 28679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7413  df-trkgcb 28695  df-trkg 28698

This theorem is referenced by:  tgcgrtriv  28729  tgbtwntriv2  28732  tgbtwnouttr2  28740  tgbtwndiff  28751  tgifscgr  28753  tgcgrxfr  28763  lnext  28812  tgbtwnconn1lem3  28819  tgbtwnconn1  28820  legtrid  28836  hlcgrex  28864  mirreu3  28907  miriso  28923  midexlem  28945  footexALT  28973  footex  28976  opphllem  28991  flatcgra  29108  dfcgra2  29114  f1otrg  29186