Theorem axtgsegcon 28548

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 28537	. . . . . 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 4192	. . . . . . 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 4191	. . . . . . 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 3933	. . . 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 28540	. . . . . 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 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 7375	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
18	17	eleq2d 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
19	18	anbi1d 632	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
20	19	rexbidv 3162	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏))))
21	20	2ralbidv 3202	. . . . 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 2825	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
23		oveq1 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑧) = (𝑌 − 𝑧))
24	23	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑦 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝑎 − 𝑏)))
25	22, 24	anbi12d 633	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
26	25	rexbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝑃 (𝑦 ∈ (𝑋𝐼𝑧) ∧ (𝑦 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏))))
27	26	2ralbidv 3202	. . . . 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 3589	. . . 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 585	. . 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 7375	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏))
34	33	eqeq2d 2748	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑌 − 𝑧) = (𝑎 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝑏)))
35	34	anbi2d 631	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
36	35	rexbidv 3162	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝑎 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏))))
37		oveq2 7376	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵))
38	37	eqeq2d 2748	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑌 − 𝑧) = (𝐴 − 𝑏) ↔ (𝑌 − 𝑧) = (𝐴 − 𝐵)))
39	38	anbi2d 631	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
40	39	rexbidv 3162	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝑏)) ↔ ∃𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵))))
41	36, 40	rspc2v 3589	. . 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 585	. 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 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442  [wsbc 3742   ∖ cdif 3900   ∩ cin 3902  {csn 4582  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  distcds 17198  TarskiGcstrkg 28511  TarskiG_Ccstrkgc 28512  TarskiG_Bcstrkgb 28513  TarskiG_CBcstrkgcb 28514  Itvcitv 28517  LineGclng 28518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-trkgcb 28534  df-trkg 28537

This theorem is referenced by:  tgcgrtriv  28568  tgbtwntriv2  28571  tgbtwnouttr2  28579  tgbtwndiff  28590  tgifscgr  28592  tgcgrxfr  28602  lnext  28651  tgbtwnconn1lem3  28658  tgbtwnconn1  28659  legtrid  28675  hlcgrex  28700  mirreu3  28738  miriso  28754  midexlem  28776  footexALT  28802  footex  28805  opphllem  28819  flatcgra  28908  dfcgra2  28914  f1otrg  28955