Metamath Proof Explorer


Theorem axtgsegcon

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 A B , 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)

Ref Expression
Hypotheses 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 axtgsegcon ( 𝜑 → ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 axtrkg.p 𝑃 = ( Base ‘ 𝐺 )
2 axtrkg.d = ( dist ‘ 𝐺 )
3 axtrkg.i 𝐼 = ( Itv ‘ 𝐺 )
4 axtrkg.g ( 𝜑𝐺 ∈ TarskiG )
5 axtgsegcon.1 ( 𝜑𝑋𝑃 )
6 axtgsegcon.2 ( 𝜑𝑌𝑃 )
7 axtgsegcon.3 ( 𝜑𝐴𝑃 )
8 axtgsegcon.4 ( 𝜑𝐵𝑃 )
9 df-trkg TarskiG = ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓[ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) )
10 inss2 ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓[ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ) ⊆ ( TarskiGCB ∩ { 𝑓[ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } )
11 inss1 ( TarskiGCB ∩ { 𝑓[ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ⊆ TarskiGCB
12 10 11 sstri ( ( TarskiGC ∩ TarskiGB ) ∩ ( TarskiGCB ∩ { 𝑓[ ( Base ‘ 𝑓 ) / 𝑝 ] [ ( Itv ‘ 𝑓 ) / 𝑖 ] ( LineG ‘ 𝑓 ) = ( 𝑥𝑝 , 𝑦 ∈ ( 𝑝 ∖ { 𝑥 } ) ↦ { 𝑧𝑝 ∣ ( 𝑧 ∈ ( 𝑥 𝑖 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝑖 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝑖 𝑧 ) ) } ) } ) ) ⊆ TarskiGCB
13 9 12 eqsstri TarskiG ⊆ TarskiGCB
14 13 4 sseldi ( 𝜑𝐺 ∈ TarskiGCB )
15 1 2 3 istrkgcb ( 𝐺 ∈ TarskiGCB ↔ ( 𝐺 ∈ V ∧ ( ∀ 𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 ( ( ( 𝑥𝑦𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ 𝑏 ∈ ( 𝑎 𝐼 𝑐 ) ) ∧ ( ( ( 𝑥 𝑦 ) = ( 𝑎 𝑏 ) ∧ ( 𝑦 𝑧 ) = ( 𝑏 𝑐 ) ) ∧ ( ( 𝑥 𝑢 ) = ( 𝑎 𝑣 ) ∧ ( 𝑦 𝑢 ) = ( 𝑏 𝑣 ) ) ) ) → ( 𝑧 𝑢 ) = ( 𝑐 𝑣 ) ) ∧ ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ) ) )
16 15 simprbi ( 𝐺 ∈ TarskiGCB → ( ∀ 𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 ( ( ( 𝑥𝑦𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ 𝑏 ∈ ( 𝑎 𝐼 𝑐 ) ) ∧ ( ( ( 𝑥 𝑦 ) = ( 𝑎 𝑏 ) ∧ ( 𝑦 𝑧 ) = ( 𝑏 𝑐 ) ) ∧ ( ( 𝑥 𝑢 ) = ( 𝑎 𝑣 ) ∧ ( 𝑦 𝑢 ) = ( 𝑏 𝑣 ) ) ) ) → ( 𝑧 𝑢 ) = ( 𝑐 𝑣 ) ) ∧ ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
17 16 simprd ( 𝐺 ∈ TarskiGCB → ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) )
18 14 17 syl ( 𝜑 → ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) )
19 oveq1 ( 𝑥 = 𝑋 → ( 𝑥 𝐼 𝑧 ) = ( 𝑋 𝐼 𝑧 ) )
20 19 eleq2d ( 𝑥 = 𝑋 → ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ↔ 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ) )
21 20 anbi1d ( 𝑥 = 𝑋 → ( ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
22 21 rexbidv ( 𝑥 = 𝑋 → ( ∃ 𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ∃ 𝑧𝑃 ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
23 22 2ralbidv ( 𝑥 = 𝑋 → ( ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
24 eleq1 ( 𝑦 = 𝑌 → ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ↔ 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ) )
25 oveq1 ( 𝑦 = 𝑌 → ( 𝑦 𝑧 ) = ( 𝑌 𝑧 ) )
26 25 eqeq1d ( 𝑦 = 𝑌 → ( ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ↔ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) )
27 24 26 anbi12d ( 𝑦 = 𝑌 → ( ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
28 27 rexbidv ( 𝑦 = 𝑌 → ( ∃ 𝑧𝑃 ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
29 28 2ralbidv ( 𝑦 = 𝑌 → ( ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
30 23 29 rspc2v ( ( 𝑋𝑃𝑌𝑃 ) → ( ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) → ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
31 5 6 30 syl2anc ( 𝜑 → ( ∀ 𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ∧ ( 𝑦 𝑧 ) = ( 𝑎 𝑏 ) ) → ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ) )
32 18 31 mpd ( 𝜑 → ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) )
33 oveq1 ( 𝑎 = 𝐴 → ( 𝑎 𝑏 ) = ( 𝐴 𝑏 ) )
34 33 eqeq2d ( 𝑎 = 𝐴 → ( ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ↔ ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ) )
35 34 anbi2d ( 𝑎 = 𝐴 → ( ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ) ) )
36 35 rexbidv ( 𝑎 = 𝐴 → ( ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) ↔ ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ) ) )
37 oveq2 ( 𝑏 = 𝐵 → ( 𝐴 𝑏 ) = ( 𝐴 𝐵 ) )
38 37 eqeq2d ( 𝑏 = 𝐵 → ( ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ↔ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) )
39 38 anbi2d ( 𝑏 = 𝐵 → ( ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ) ↔ ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) ) )
40 39 rexbidv ( 𝑏 = 𝐵 → ( ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝑏 ) ) ↔ ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) ) )
41 36 40 rspc2v ( ( 𝐴𝑃𝐵𝑃 ) → ( ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) → ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) ) )
42 7 8 41 syl2anc ( 𝜑 → ( ∀ 𝑎𝑃𝑏𝑃𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝑎 𝑏 ) ) → ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) ) )
43 32 42 mpd ( 𝜑 → ∃ 𝑧𝑃 ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ∧ ( 𝑌 𝑧 ) = ( 𝐴 𝐵 ) ) )