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Theorem legval 25678
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 3352 . . 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 1131 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
87eqcomd 2766 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
9 simp2 1132 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
109eqcomd 2766 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
1110oveqd 6830 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑦) = (𝑥𝑑𝑦))
1211eqeq2d 2770 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑓 = (𝑥 𝑦) ↔ 𝑓 = (𝑥𝑑𝑦)))
13 simp3 1133 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
1413eqcomd 2766 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
1514oveqd 6830 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
1615eleq2d 2825 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
1710oveqd 6830 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑧) = (𝑥𝑑𝑧))
1817eqeq2d 2770 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥𝑑𝑧)))
1916, 18anbi12d 749 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
208, 19rexeqbidv 3292 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
2112, 20anbi12d 749 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) ↔ (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
228, 21rexeqbidv 3292 . . . . . . 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 3292 . . . . . 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 16119 . . . . 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 4868 . . . 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 25677 . . . 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 6362 . . . . . . . . . 10 (dist‘𝐺) ∈ V
285, 27eqeltri 2835 . . . . . . . . 9 ∈ V
2928imaex 7269 . . . . . . . 8 ( “ (𝑃 × 𝑃)) ∈ V
30 p0ex 5002 . . . . . . . 8 {∅} ∈ V
3129, 30unex 7121 . . . . . . 7 (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V
3231a1i 11 . . . . . 6 (⊤ → (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V)
33 simprr 813 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 = (𝑥 𝑑))
34 ovima0 6978 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑑𝑃) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3534ad5ant14 1218 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3633, 35eqeltrd 2839 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
37 simpllr 817 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 = (𝑥 𝑦))
39 ovima0 6978 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑦𝑃) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4039ad3antrrr 768 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4138, 40eqeltrd 2839 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4236, 41jca 555 . . . . . . . . . . 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 813 . . . . . . . . . . . 12 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))
44 eleq1w 2822 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑑 ∈ (𝑥𝐼𝑦)))
45 oveq2 6821 . . . . . . . . . . . . . . 15 (𝑧 = 𝑑 → (𝑥 𝑧) = (𝑥 𝑑))
4645eqeq2d 2770 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥 𝑑)))
4744, 46anbi12d 749 . . . . . . . . . . . . 13 (𝑧 = 𝑑 → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))))
4847cbvrexv 3311 . . . . . . . . . . . 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 3216 . . . . . . . . . 10 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5150ex 449 . . . . . . . . 9 ((𝑥𝑃𝑦𝑃) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))))
5251rexlimivv 3174 . . . . . . . 8 (∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5352adantl 473 . . . . . . 7 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5453simpld 477 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5553simprd 482 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5632, 32, 54, 55opabex2 7394 . . . . 5 (⊤ → {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V)
5756trud 1642 . . . 4 {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V
5825, 26, 57fvmpt 6444 . . 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 2806 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 383  w3a 1072   = wceq 1632  wtru 1633  wcel 2139  wrex 3051  Vcvv 3340  [wsbc 3576  cun 3713  c0 4058  {csn 4321  {copab 4864   × cxp 5264  cima 5269  cfv 6049  (class class class)co 6813  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  Itvcitv 25534  ≤Gcleg 25676
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-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
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-mo 2612  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-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-leg 25677
This theorem is referenced by:  legov  25679
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