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Theorem legval 28607
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 3499 . . 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 1135 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
87eqcomd 2741 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
9 simp2 1136 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
109eqcomd 2741 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
1110oveqd 7448 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑦) = (𝑥𝑑𝑦))
1211eqeq2d 2746 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑓 = (𝑥 𝑦) ↔ 𝑓 = (𝑥𝑑𝑦)))
13 simp3 1137 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
1413eqcomd 2741 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
1514oveqd 7448 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
1615eleq2d 2825 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
1710oveqd 7448 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑧) = (𝑥𝑑𝑧))
1817eqeq2d 2746 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥𝑑𝑧)))
1916, 18anbi12d 632 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
208, 19rexeqbidv 3345 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
2112, 20anbi12d 632 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) ↔ (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
228, 21rexeqbidv 3345 . . . . . . 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 3345 . . . . . 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 17196 . . . . 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 5214 . . . 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 28606 . . . 4 ≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
275fvexi 6921 . . . . . . . . 9 ∈ V
2827imaex 7937 . . . . . . . 8 ( “ (𝑃 × 𝑃)) ∈ V
29 p0ex 5390 . . . . . . . 8 {∅} ∈ V
3028, 29unex 7763 . . . . . . 7 (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V
3130a1i 11 . . . . . 6 (⊤ → (( “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V)
32 simprr 773 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 = (𝑥 𝑑))
33 ovima0 7612 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑑𝑃) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3433ad5ant14 758 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑑) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3532, 34eqeltrd 2839 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
36 simpllr 776 . . . . . . . . . . . . . 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))))
3736simpld 494 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 = (𝑥 𝑦))
38 ovima0 7612 . . . . . . . . . . . . . 14 ((𝑥𝑃𝑦𝑃) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
3938ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑥 𝑦) ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4037, 39eqeltrd 2839 . . . . . . . . . . . 12 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
4135, 40jca 511 . . . . . . . . . . 11 (((((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) ∧ 𝑑𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
42 simprr 773 . . . . . . . . . . . 12 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))
43 eleq1w 2822 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑑 ∈ (𝑥𝐼𝑦)))
44 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑧 = 𝑑 → (𝑥 𝑧) = (𝑥 𝑑))
4544eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑧 = 𝑑 → (𝑒 = (𝑥 𝑧) ↔ 𝑒 = (𝑥 𝑑)))
4643, 45anbi12d 632 . . . . . . . . . . . . 13 (𝑧 = 𝑑 → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑))))
4746cbvrexvw 3236 . . . . . . . . . . . 12 (∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)) ↔ ∃𝑑𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑)))
4842, 47sylib 218 . . . . . . . . . . 11 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → ∃𝑑𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑑)))
4941, 48r19.29a 3160 . . . . . . . . . 10 (((𝑥𝑃𝑦𝑃) ∧ (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5049ex 412 . . . . . . . . 9 ((𝑥𝑃𝑦𝑃) → ((𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))))
5150rexlimivv 3199 . . . . . . . 8 (∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5251adantl 481 . . . . . . 7 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → (𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅})))
5352simpld 494 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑒 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5452simprd 495 . . . . . 6 ((⊤ ∧ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))) → 𝑓 ∈ (( “ (𝑃 × 𝑃)) ∪ {∅}))
5531, 31, 53, 54opabex2 8081 . . . . 5 (⊤ → {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V)
5655mptru 1544 . . . 4 {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))} ∈ V
5725, 26, 56fvmpt 7016 . . 3 (𝐺 ∈ V → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
582, 3, 573syl 18 . 2 (𝜑 → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
591, 58eqtrid 2787 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 395  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  wrex 3068  Vcvv 3478  [wsbc 3791  cun 3961  c0 4339  {csn 4631  {copab 5210   × cxp 5687  cima 5692  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  ≤Gcleg 28605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-leg 28606
This theorem is referenced by:  legov  28608
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