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Theorem legval 25379
Description: Value of the less-than relationship. (Contributed by Thierry Arnoux, 21-Jun-2019.)
Hypotheses
Ref Expression
legval.p 𝑃 = (Base‘𝐺)
legval.d = (dist‘𝐺)
legval.i 𝐼 = (Itv‘𝐺)
legval.l = (≤G‘𝐺)
legval.g (𝜑𝐺 ∈ TarskiG)
Assertion
Ref Expression
legval (𝜑 = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
Distinct variable groups:   𝑒,𝑓,𝐺   𝑥,𝑦,𝑧,𝐼   𝑥,𝑒,𝑦,𝑧,𝑃,𝑓   ,𝑒,𝑓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑒,𝑓)   𝐺(𝑥,𝑦,𝑧)   𝐼(𝑒,𝑓)   (𝑥,𝑦,𝑧,𝑒,𝑓)

Proof of Theorem legval
Dummy variables 𝑑 𝑔 𝑖 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 legval.l . 2 = (≤G‘𝐺)
2 legval.g . . 3 (𝜑𝐺 ∈ TarskiG)
3 elex 3198 . . 3 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
4 legval.p . . . . . 6 𝑃 = (Base‘𝐺)
5 legval.d . . . . . 6 = (dist‘𝐺)
6 legval.i . . . . . 6 𝐼 = (Itv‘𝐺)
7 simp1 1059 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
87eqcomd 2627 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
9 simp2 1060 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
109eqcomd 2627 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
1110oveqd 6621 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑦) = (𝑥𝑑𝑦))
1211eqeq2d 2631 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑓 = (𝑥 𝑦) ↔ 𝑓 = (𝑥𝑑𝑦)))
13 simp3 1061 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
1413eqcomd 2627 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
1514oveqd 6621 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
1615eleq2d 2684 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
1710oveqd 6621 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑧) = (𝑥𝑑𝑧))
1817eqeq2d 2631 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥𝑑𝑧)))
1916, 18anbi12d 746 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
208, 19rexeqbidv 3142 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
2112, 20anbi12d 746 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) ↔ (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
228, 21rexeqbidv 3142 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) ↔ ∃𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
238, 22rexeqbidv 3142 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) ↔ ∃𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
244, 5, 6, 23sbcie3s 15838 . . . . 5 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))) ↔ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))))
2524opabbidv 4678 . . . 4 (𝑔 = 𝐺 → {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))} = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
26 df-leg 25378 . . . 4 ≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
27 fvex 6158 . . . . . . . . . 10 (dist‘𝐺) ∈ V
285, 27eqeltri 2694 . . . . . . . . 9 ∈ V
2928imaex 7051 . . . . . . . 8 ( “ (𝑃 × 𝑃)) ∈ V
30 p0ex 4813 . . . . . . . 8 {∅} ∈ V
3129, 30unex 6909 . . . . . . 7 (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V
3231a1i 11 . . . . . 6 (⊤ → (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V)
33 simprr 795 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 = (𝑥 𝑑))
34 ovima0 6766 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑑𝑃) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3534ad5ant14 1299 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3633, 35eqeltrd 2698 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
37 simpllr 798 . . . . . . . . . . . . . 14 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))))
3837simpld 475 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 = (𝑥 𝑦))
39 ovima0 6766 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑦𝑃) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4039ad3antrrr 765 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4138, 40eqeltrd 2698 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4236, 41jca 554 . . . . . . . . . . 11 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
43 simprr 795 . . . . . . . . . . . 12 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))
44 eleq1 2686 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑑 ∈ (𝑥𝐼𝑦)))
45 oveq2 6612 . . . . . . . . . . . . . . 15 (𝑧 = 𝑑 → (𝑥 𝑧) = (𝑥 𝑑))
4645eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥 𝑑)))
4744, 46anbi12d 746 . . . . . . . . . . . . 13 (𝑧 = 𝑑 → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))))
4847cbvrexv 3160 . . . . . . . . . . . 12 (∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ ∃𝑑𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑)))
4943, 48sylib 208 . . . . . . . . . . 11 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → ∃𝑑𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑)))
5042, 49r19.29a 3071 . . . . . . . . . 10 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5150ex 450 . . . . . . . . 9 ((𝑥𝑃𝑦𝑃) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))))
5251rexlimivv 3029 . . . . . . . 8 (∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5352adantl 482 . . . . . . 7 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5453simpld 475 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5553simprd 479 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5632, 32, 54, 55opabex2 7172 . . . . 5 (⊤ → {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V)
5756trud 1490 . . . 4 {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V
5825, 26, 57fvmpt 6239 . . 3 (𝐺 ∈ V → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
592, 3, 583syl 18 . 2 (𝜑 → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
601, 59syl5eq 2667 1 (𝜑 = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  wrex 2908  Vcvv 3186  [wsbc 3417  cun 3553  c0 3891  {csn 4148  {copab 4672   × cxp 5072  cima 5077  cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235  ≤Gcleg 25377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-leg 25378
This theorem is referenced by:  legov  25380
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