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Theorem ltrneq2 38611
Description: The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
Hypotheses
Ref Expression
ltrneq2.a 𝐴 = (Atoms‘𝐾)
ltrneq2.h 𝐻 = (LHyp‘𝐾)
ltrneq2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrneq2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ↔ 𝐹 = 𝐺))
Distinct variable groups:   𝐴,𝑝   𝐹,𝑝   𝐺,𝑝
Allowed substitution hints:   𝑇(𝑝)   𝐻(𝑝)   𝐾(𝑝)   𝑊(𝑝)

Proof of Theorem ltrneq2
Dummy variables 𝑞 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1191 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1193 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺𝑇)
3 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝐾) = (Base‘𝐾)
4 ltrneq2.h . . . . . . . . . . . . . . 15 𝐻 = (LHyp‘𝐾)
5 ltrneq2.t . . . . . . . . . . . . . . 15 𝑇 = ((LTrn‘𝐾)‘𝑊)
63, 4, 5ltrn1o 38587 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
71, 2, 6syl2anc 584 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8 simpl2 1192 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹𝑇)
9 simpr3 1196 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞𝐴)
10 eqid 2736 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
11 ltrneq2.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
1210, 11, 4, 5ltrncnvat 38604 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞𝐴) → (𝐹𝑞) ∈ 𝐴)
131, 8, 9, 12syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) ∈ 𝐴)
143, 11atbase 37751 . . . . . . . . . . . . . 14 ((𝐹𝑞) ∈ 𝐴 → (𝐹𝑞) ∈ (Base‘𝐾))
1513, 14syl 17 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) ∈ (Base‘𝐾))
16 f1ocnvfv1 7222 . . . . . . . . . . . . 13 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (𝐹𝑞) ∈ (Base‘𝐾)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
177, 15, 16syl2anc 584 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
18 simpr2 1195 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
19 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐹𝑝) = (𝐹‘(𝐹𝑞)))
20 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐺𝑝) = (𝐺‘(𝐹𝑞)))
2119, 20eqeq12d 2752 . . . . . . . . . . . . . . . 16 (𝑝 = (𝐹𝑞) → ((𝐹𝑝) = (𝐺𝑝) ↔ (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞))))
2221rspcv 3577 . . . . . . . . . . . . . . 15 ((𝐹𝑞) ∈ 𝐴 → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞))))
2313, 18, 22sylc 65 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞)))
243, 4, 5ltrn1o 38587 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
251, 8, 24syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
263, 11atbase 37751 . . . . . . . . . . . . . . . 16 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
279, 26syl 17 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
28 f1ocnvfv2 7223 . . . . . . . . . . . . . . 15 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
2925, 27, 28syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑞)) = 𝑞)
3023, 29eqtr3d 2778 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐹𝑞)) = 𝑞)
3130fveq2d 6846 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐺𝑞))
3217, 31eqtr3d 2778 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) = (𝐺𝑞))
3332breq1d 5115 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)𝑥 ↔ (𝐺𝑞)(le‘𝐾)𝑥))
34 simpr1 1194 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑥 ∈ (Base‘𝐾))
35 f1ocnvfv1 7222 . . . . . . . . . . . 12 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3625, 34, 35syl2anc 584 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3736breq2d 5117 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑞)(le‘𝐾)𝑥))
38 f1ocnvfv1 7222 . . . . . . . . . . . 12 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘(𝐺𝑥)) = 𝑥)
397, 34, 38syl2anc 584 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺𝑥)) = 𝑥)
4039breq2d 5117 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥)) ↔ (𝐺𝑞)(le‘𝐾)𝑥))
4133, 37, 403bitr4d 310 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
42 simpl1l 1224 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐾 ∈ HL)
43 eqid 2736 . . . . . . . . . . . 12 (LAut‘𝐾) = (LAut‘𝐾)
444, 43, 5ltrnlaut 38586 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ (LAut‘𝐾))
451, 8, 44syl2anc 584 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹 ∈ (LAut‘𝐾))
463, 4, 5ltrncl 38588 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
471, 8, 34, 46syl3anc 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑥) ∈ (Base‘𝐾))
483, 10, 43lautcnvle 38552 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ (𝐹𝑥) ∈ (Base‘𝐾))) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ (𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥))))
4942, 45, 27, 47, 48syl22anc 837 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ (𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥))))
504, 43, 5ltrnlaut 38586 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺 ∈ (LAut‘𝐾))
511, 2, 50syl2anc 584 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺 ∈ (LAut‘𝐾))
523, 4, 5ltrncl 38588 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
531, 2, 34, 52syl3anc 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺𝑥) ∈ (Base‘𝐾))
543, 10, 43lautcnvle 38552 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝐺 ∈ (LAut‘𝐾)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ (𝐺𝑥) ∈ (Base‘𝐾))) → (𝑞(le‘𝐾)(𝐺𝑥) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
5542, 51, 27, 53, 54syl22anc 837 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐺𝑥) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
5641, 49, 553bitr4d 310 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))
57563exp2 1354 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝑥 ∈ (Base‘𝐾) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))))
5857imp 407 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))))
5958ralrimdv 3149 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))
60 simpl1l 1224 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61 simpl1 1191 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
62 simpl2 1192 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹𝑇)
63 simpr 485 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
6461, 62, 63, 46syl3anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
65 simpl3 1193 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺𝑇)
6661, 65, 63, 52syl3anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
673, 10, 11hlateq 37862 . . . . . 6 ((𝐾 ∈ HL ∧ (𝐹𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑥) ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6860, 64, 66, 67syl3anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6959, 68sylibd 238 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝐹𝑥) = (𝐺𝑥)))
7069ralrimdva 3151 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
71243adant3 1132 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72 f1ofn 6785 . . . . 5 (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹 Fn (Base‘𝐾))
7371, 72syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐹 Fn (Base‘𝐾))
7463adant2 1131 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75 f1ofn 6785 . . . . 5 (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺 Fn (Base‘𝐾))
7674, 75syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐺 Fn (Base‘𝐾))
77 eqfnfv 6982 . . . 4 ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7873, 76, 77syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7970, 78sylibrd 258 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → 𝐹 = 𝐺))
80 fveq1 6841 . . 3 (𝐹 = 𝐺 → (𝐹𝑝) = (𝐺𝑝))
8180ralrimivw 3147 . 2 (𝐹 = 𝐺 → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
8279, 81impbid1 224 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105  ccnv 5632   Fn wfn 6491  1-1-ontowf1o 6495  cfv 6496  Basecbs 17083  lecple 17140  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LAutclaut 38448  LTrncltrn 38564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568
This theorem is referenced by:  ltrneq  38612  cdlemd  38670
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