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Theorem axtg5seg 25563
 Description: Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles 𝑋𝑍𝑈 and 𝐴𝐶𝑉, a point 𝑌 on 𝑋𝑍, and a point 𝐵 on 𝐴𝐶. If all corresponding line segments except for 𝑍𝑈 and 𝐶𝑉 are congruent ( i.e., 𝑋𝑌 ∼ 𝐴𝐵, 𝑌𝑍 ∼ 𝐵𝐶, 𝑋𝑈 ∼ 𝐴𝑉, and 𝑌𝑈 ∼ 𝐵𝑉), then 𝑍𝑈 and 𝐶𝑉 are also congruent. As noted in Axiom 5 of [Tarski1999] p. 178, "this axiom is similar in character to the well-known theorems of Euclidean geometry that allow one to conclude, from hypotheses about the congruence of certain corresponding sides and angles in two triangles, the congruence of other corresponding sides and angles." (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Base‘𝐺)
axtrkg.d = (dist‘𝐺)
axtrkg.i 𝐼 = (Itv‘𝐺)
axtrkg.g (𝜑𝐺 ∈ TarskiG)
axtg5seg.1 (𝜑𝑋𝑃)
axtg5seg.2 (𝜑𝑌𝑃)
axtg5seg.3 (𝜑𝑍𝑃)
axtg5seg.4 (𝜑𝐴𝑃)
axtg5seg.5 (𝜑𝐵𝑃)
axtg5seg.6 (𝜑𝐶𝑃)
axtg5seg.7 (𝜑𝑈𝑃)
axtg5seg.8 (𝜑𝑉𝑃)
axtg5seg.9 (𝜑𝑋𝑌)
axtg5seg.10 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
axtg5seg.11 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
axtg5seg.12 (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))
axtg5seg.13 (𝜑 → (𝑌 𝑍) = (𝐵 𝐶))
axtg5seg.14 (𝜑 → (𝑋 𝑈) = (𝐴 𝑉))
axtg5seg.15 (𝜑 → (𝑌 𝑈) = (𝐵 𝑉))
Assertion
Ref Expression
axtg5seg (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))

Proof of Theorem axtg5seg
Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 25551 . . . . . . 7 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2 inss2 3977 . . . . . . . 8 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
3 inss1 3976 . . . . . . . 8 (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}) ⊆ TarskiGCB
42, 3sstri 3753 . . . . . . 7 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGCB
51, 4eqsstri 3776 . . . . . 6 TarskiG ⊆ TarskiGCB
6 axtrkg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3742 . . . . 5 (𝜑𝐺 ∈ TarskiGCB)
8 axtrkg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
9 axtrkg.d . . . . . . . 8 = (dist‘𝐺)
10 axtrkg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
118, 9, 10istrkgcb 25554 . . . . . . 7 (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))))
1211simprbi 483 . . . . . 6 (𝐺 ∈ TarskiGCB → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏))))
1312simpld 477 . . . . 5 (𝐺 ∈ TarskiGCB → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)))
147, 13syl 17 . . . 4 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)))
15 axtg5seg.1 . . . . 5 (𝜑𝑋𝑃)
16 axtg5seg.2 . . . . 5 (𝜑𝑌𝑃)
17 axtg5seg.3 . . . . 5 (𝜑𝑍𝑃)
18 neeq1 2994 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
19 oveq1 6820 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2019eleq2d 2825 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2118, 203anbi12d 1549 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
22 oveq1 6820 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2322eqeq1d 2762 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑦) = (𝑎 𝑏)))
2423anbi1d 743 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐))))
25 oveq1 6820 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑥 𝑢) = (𝑋 𝑢))
2625eqeq1d 2762 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑥 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑢) = (𝑎 𝑣)))
2726anbi1d 743 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))))
2824, 27anbi12d 749 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))))
2921, 28anbi12d 749 . . . . . . . . . 10 (𝑥 = 𝑋 → (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))))))
3029imbi1d 330 . . . . . . . . 9 (𝑥 = 𝑋 → ((((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
3130ralbidv 3124 . . . . . . . 8 (𝑥 = 𝑋 → (∀𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
32312ralbidv 3127 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
33322ralbidv 3127 . . . . . 6 (𝑥 = 𝑋 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
34 neeq2 2995 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝑦𝑋𝑌))
35 eleq1 2827 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
3634, 353anbi12d 1549 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
37 oveq2 6821 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
3837eqeq1d 2762 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝑎 𝑏)))
39 oveq1 6820 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
4039eqeq1d 2762 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑧) = (𝑏 𝑐)))
4138, 40anbi12d 749 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐))))
42 oveq1 6820 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑦 𝑢) = (𝑌 𝑢))
4342eqeq1d 2762 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑦 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑢) = (𝑏 𝑣)))
4443anbi2d 742 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))
4541, 44anbi12d 749 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))))
4636, 45anbi12d 749 . . . . . . . . . 10 (𝑦 = 𝑌 → (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))))
4746imbi1d 330 . . . . . . . . 9 (𝑦 = 𝑌 → ((((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
4847ralbidv 3124 . . . . . . . 8 (𝑦 = 𝑌 → (∀𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
49482ralbidv 3127 . . . . . . 7 (𝑦 = 𝑌 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
50492ralbidv 3127 . . . . . 6 (𝑦 = 𝑌 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑦𝑦 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣))))
51 oveq2 6821 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5251eleq2d 2825 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
53523anbi2d 1553 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐))))
54 oveq2 6821 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
5554eqeq1d 2762 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → ((𝑌 𝑧) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝑏 𝑐)))
5655anbi2d 742 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
5756anbi1d 743 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))))
5853, 57anbi12d 749 . . . . . . . . . 10 (𝑧 = 𝑍 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))))))
59 oveq1 6820 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 𝑢) = (𝑍 𝑢))
6059eqeq1d 2762 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑢) = (𝑐 𝑣)))
6158, 60imbi12d 333 . . . . . . . . 9 (𝑧 = 𝑍 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6261ralbidv 3124 . . . . . . . 8 (𝑧 = 𝑍 → (∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
63622ralbidv 3127 . . . . . . 7 (𝑧 = 𝑍 → (∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
64632ralbidv 3127 . . . . . 6 (𝑧 = 𝑍 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑧) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ↔ ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6533, 50, 64rspc3v 3464 . . . . 5 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6615, 16, 17, 65syl3anc 1477 . . . 4 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣))))
6714, 66mpd 15 . . 3 (𝜑 → ∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)))
68 axtg5seg.7 . . . 4 (𝜑𝑈𝑃)
69 axtg5seg.4 . . . 4 (𝜑𝐴𝑃)
70 axtg5seg.5 . . . 4 (𝜑𝐵𝑃)
71 oveq2 6821 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑋 𝑢) = (𝑋 𝑈))
7271eqeq1d 2762 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑋 𝑢) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝑎 𝑣)))
73 oveq2 6821 . . . . . . . . . . 11 (𝑢 = 𝑈 → (𝑌 𝑢) = (𝑌 𝑈))
7473eqeq1d 2762 . . . . . . . . . 10 (𝑢 = 𝑈 → ((𝑌 𝑢) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝑏 𝑣)))
7572, 74anbi12d 749 . . . . . . . . 9 (𝑢 = 𝑈 → (((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
7675anbi2d 742 . . . . . . . 8 (𝑢 = 𝑈 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))))
7776anbi2d 742 . . . . . . 7 (𝑢 = 𝑈 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))))
78 oveq2 6821 . . . . . . . 8 (𝑢 = 𝑈 → (𝑍 𝑢) = (𝑍 𝑈))
7978eqeq1d 2762 . . . . . . 7 (𝑢 = 𝑈 → ((𝑍 𝑢) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝑐 𝑣)))
8077, 79imbi12d 333 . . . . . 6 (𝑢 = 𝑈 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
81802ralbidv 3127 . . . . 5 (𝑢 = 𝑈 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
82 oveq1 6820 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝐼𝑐) = (𝐴𝐼𝑐))
8382eleq2d 2825 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑏 ∈ (𝑎𝐼𝑐) ↔ 𝑏 ∈ (𝐴𝐼𝑐)))
84833anbi3d 1554 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐))))
85 oveq1 6820 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑏) = (𝐴 𝑏))
8685eqeq2d 2770 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑌) = (𝑎 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝑏)))
8786anbi1d 743 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐))))
88 oveq1 6820 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎 𝑣) = (𝐴 𝑣))
8988eqeq2d 2770 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑋 𝑈) = (𝑎 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑣)))
9089anbi1d 743 . . . . . . . . 9 (𝑎 = 𝐴 → (((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))
9187, 90anbi12d 749 . . . . . . . 8 (𝑎 = 𝐴 → ((((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))))
9284, 91anbi12d 749 . . . . . . 7 (𝑎 = 𝐴 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))))))
9392imbi1d 330 . . . . . 6 (𝑎 = 𝐴 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
94932ralbidv 3127 . . . . 5 (𝑎 = 𝐴 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝑎 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
95 eleq1 2827 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝑐)))
96953anbi3d 1554 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐))))
97 oveq2 6821 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴 𝑏) = (𝐴 𝐵))
9897eqeq2d 2770 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑋 𝑌) = (𝐴 𝑏) ↔ (𝑋 𝑌) = (𝐴 𝐵)))
99 oveq1 6820 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑐) = (𝐵 𝑐))
10099eqeq2d 2770 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑍) = (𝑏 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝑐)))
10198, 100anbi12d 749 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐))))
102 oveq1 6820 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏 𝑣) = (𝐵 𝑣))
103102eqeq2d 2770 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑌 𝑈) = (𝑏 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑣)))
104103anbi2d 742 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))
105101, 104anbi12d 749 . . . . . . . 8 (𝑏 = 𝐵 → ((((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))))
10696, 105anbi12d 749 . . . . . . 7 (𝑏 = 𝐵 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))))
107106imbi1d 330 . . . . . 6 (𝑏 = 𝐵 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
1081072ralbidv 3127 . . . . 5 (𝑏 = 𝐵 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝑏 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
10981, 94, 108rspc3v 3464 . . . 4 ((𝑈𝑃𝐴𝑃𝐵𝑃) → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
11068, 69, 70, 109syl3anc 1477 . . 3 (𝜑 → (∀𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝑎 𝑏) ∧ (𝑌 𝑍) = (𝑏 𝑐)) ∧ ((𝑋 𝑢) = (𝑎 𝑣) ∧ (𝑌 𝑢) = (𝑏 𝑣)))) → (𝑍 𝑢) = (𝑐 𝑣)) → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣))))
11167, 110mpd 15 . 2 (𝜑 → ∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)))
112 axtg5seg.9 . . . 4 (𝜑𝑋𝑌)
113 axtg5seg.10 . . . 4 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
114 axtg5seg.11 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
115112, 113, 1143jca 1123 . . 3 (𝜑 → (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)))
116 axtg5seg.12 . . . 4 (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))
117 axtg5seg.13 . . . 4 (𝜑 → (𝑌 𝑍) = (𝐵 𝐶))
118116, 117jca 555 . . 3 (𝜑 → ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)))
119 axtg5seg.14 . . . 4 (𝜑 → (𝑋 𝑈) = (𝐴 𝑉))
120 axtg5seg.15 . . . 4 (𝜑 → (𝑌 𝑈) = (𝐵 𝑉))
121119, 120jca 555 . . 3 (𝜑 → ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))
122115, 118, 121jca32 559 . 2 (𝜑 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))))
123 axtg5seg.6 . . 3 (𝜑𝐶𝑃)
124 axtg5seg.8 . . 3 (𝜑𝑉𝑃)
125 oveq2 6821 . . . . . . . 8 (𝑐 = 𝐶 → (𝐴𝐼𝑐) = (𝐴𝐼𝐶))
126125eleq2d 2825 . . . . . . 7 (𝑐 = 𝐶 → (𝐵 ∈ (𝐴𝐼𝑐) ↔ 𝐵 ∈ (𝐴𝐼𝐶)))
1271263anbi3d 1554 . . . . . 6 (𝑐 = 𝐶 → ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ↔ (𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶))))
128 oveq2 6821 . . . . . . . . 9 (𝑐 = 𝐶 → (𝐵 𝑐) = (𝐵 𝐶))
129128eqeq2d 2770 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑌 𝑍) = (𝐵 𝑐) ↔ (𝑌 𝑍) = (𝐵 𝐶)))
130129anbi2d 742 . . . . . . 7 (𝑐 = 𝐶 → (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ↔ ((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶))))
131130anbi1d 743 . . . . . 6 (𝑐 = 𝐶 → ((((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))))
132127, 131anbi12d 749 . . . . 5 (𝑐 = 𝐶 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))))))
133 oveq1 6820 . . . . . 6 (𝑐 = 𝐶 → (𝑐 𝑣) = (𝐶 𝑣))
134133eqeq2d 2770 . . . . 5 (𝑐 = 𝐶 → ((𝑍 𝑈) = (𝑐 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑣)))
135132, 134imbi12d 333 . . . 4 (𝑐 = 𝐶 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝐶 𝑣))))
136 oveq2 6821 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐴 𝑣) = (𝐴 𝑉))
137136eqeq2d 2770 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑋 𝑈) = (𝐴 𝑣) ↔ (𝑋 𝑈) = (𝐴 𝑉)))
138 oveq2 6821 . . . . . . . . 9 (𝑣 = 𝑉 → (𝐵 𝑣) = (𝐵 𝑉))
139138eqeq2d 2770 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑌 𝑈) = (𝐵 𝑣) ↔ (𝑌 𝑈) = (𝐵 𝑉)))
140137, 139anbi12d 749 . . . . . . 7 (𝑣 = 𝑉 → (((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)) ↔ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉))))
141140anbi2d 742 . . . . . 6 (𝑣 = 𝑉 → ((((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣))) ↔ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))))
142141anbi2d 742 . . . . 5 (𝑣 = 𝑉 → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) ↔ ((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉))))))
143 oveq2 6821 . . . . . 6 (𝑣 = 𝑉 → (𝐶 𝑣) = (𝐶 𝑉))
144143eqeq2d 2770 . . . . 5 (𝑣 = 𝑉 → ((𝑍 𝑈) = (𝐶 𝑣) ↔ (𝑍 𝑈) = (𝐶 𝑉)))
145142, 144imbi12d 333 . . . 4 (𝑣 = 𝑉 → ((((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝐶 𝑣)) ↔ (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
146135, 145rspc2v 3461 . . 3 ((𝐶𝑃𝑉𝑃) → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
147123, 124, 146syl2anc 696 . 2 (𝜑 → (∀𝑐𝑃𝑣𝑃 (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝑐)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝑐)) ∧ ((𝑋 𝑈) = (𝐴 𝑣) ∧ (𝑌 𝑈) = (𝐵 𝑣)))) → (𝑍 𝑈) = (𝑐 𝑣)) → (((𝑋𝑌𝑌 ∈ (𝑋𝐼𝑍) ∧ 𝐵 ∈ (𝐴𝐼𝐶)) ∧ (((𝑋 𝑌) = (𝐴 𝐵) ∧ (𝑌 𝑍) = (𝐵 𝐶)) ∧ ((𝑋 𝑈) = (𝐴 𝑉) ∧ (𝑌 𝑈) = (𝐵 𝑉)))) → (𝑍 𝑈) = (𝐶 𝑉))))
148111, 122, 147mp2d 49 1 (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  {cab 2746   ≠ wne 2932  ∀wral 3050  ∃wrex 3051  {crab 3054  Vcvv 3340  [wsbc 3576   ∖ cdif 3712   ∩ cin 3714  {csn 4321  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  TarskiGCcstrkgc 25529  TarskiGBcstrkgb 25530  TarskiGCBcstrkgcb 25531  Itvcitv 25534  LineGclng 25535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-trkgcb 25548  df-trkg 25551 This theorem is referenced by:  tgcgrextend  25579  tgsegconeq  25580  tgifscgr  25602  tgfscgr  25662  tgbtwnconn1lem2  25667  tgbtwnconn1lem3  25668  miriso  25764  midexlem  25786  ragcgr  25801  footex  25812  lmiisolem  25887  f1otrg  25950  tg5segofs  31060
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