Theorem axtgsegcon 26825

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 26814	. . . . . 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 4163	. . . . . . 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 4162	. . . . . . 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 3930	. . . . . 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 3955	. . . . 5 ⊢ TarskiG ⊆ TarskiGCB
6		axtrkg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3919	. . . 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 26817	. . . . . 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 497	. . . . 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 496	. . . 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 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
18	17	eleq2d 2824	. . . . . . . 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 3226	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
21	20	2ralbidv 3129	. . . . 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 2826	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23		oveq1 7282	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑧) = (𝑌 − 𝑧))
24	23	eqeq1d 2740	. . . . . . . 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 3226	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
27	26	2ralbidv 3129	. . . . 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 3570	. . . 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 584	. . 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 7282	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏))
34	33	eqeq2d 2749	. . . . . 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 3226	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
37		oveq2 7283	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵))
38	37	eqeq2d 2749	. . . . . 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 3226	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
41	36, 40	rspc2v 3570	. . 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 584	. 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 396   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432  [wsbc 3716   ∖ cdif 3884   ∩ cin 3886  {csn 4561  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  TarskiG_Ccstrkgc 26789  TarskiG_Bcstrkgb 26790  TarskiG_CBcstrkgcb 26791  Itvcitv 26794  LineGclng 26795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgcb 26811  df-trkg 26814

This theorem is referenced by:  tgcgrtriv  26845  tgbtwntriv2  26848  tgbtwnouttr2  26856  tgbtwndiff  26867  tgifscgr  26869  tgcgrxfr  26879  lnext  26928  tgbtwnconn1lem3  26935  tgbtwnconn1  26936  legtrid  26952  hlcgrex  26977  mirreu3  27015  miriso  27031  midexlem  27053  footexALT  27079  footex  27082  opphllem  27096  flatcgra  27185  dfcgra2  27191  f1otrg  27232