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Theorem ltrneq2 34252
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 1056 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1058 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺𝑇)
3 eqid 2606 . . . . . . . . . . . . . . 15 (Base‘𝐾) = (Base‘𝐾)
4 ltrneq2.h . . . . . . . . . . . . . . 15 𝐻 = (LHyp‘𝐾)
5 ltrneq2.t . . . . . . . . . . . . . . 15 𝑇 = ((LTrn‘𝐾)‘𝑊)
63, 4, 5ltrn1o 34228 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
71, 2, 6syl2anc 690 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8 simpl2 1057 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹𝑇)
9 simpr3 1061 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞𝐴)
10 eqid 2606 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
11 ltrneq2.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
1210, 11, 4, 5ltrncnvat 34245 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞𝐴) → (𝐹𝑞) ∈ 𝐴)
131, 8, 9, 12syl3anc 1317 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) ∈ 𝐴)
143, 11atbase 33394 . . . . . . . . . . . . . 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 6407 . . . . . . . . . . . . 13 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (𝐹𝑞) ∈ (Base‘𝐾)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
177, 15, 16syl2anc 690 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
18 simpr2 1060 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
19 fveq2 6085 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐹𝑝) = (𝐹‘(𝐹𝑞)))
20 fveq2 6085 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐺𝑝) = (𝐺‘(𝐹𝑞)))
2119, 20eqeq12d 2621 . . . . . . . . . . . . . . . 16 (𝑝 = (𝐹𝑞) → ((𝐹𝑝) = (𝐺𝑝) ↔ (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞))))
2221rspcv 3274 . . . . . . . . . . . . . . 15 ((𝐹𝑞) ∈ 𝐴 → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞))))
2313, 18, 22sylc 62 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞)))
243, 4, 5ltrn1o 34228 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
251, 8, 24syl2anc 690 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
263, 11atbase 33394 . . . . . . . . . . . . . . . 16 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
279, 26syl 17 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
28 f1ocnvfv2 6408 . . . . . . . . . . . . . . 15 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
2925, 27, 28syl2anc 690 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑞)) = 𝑞)
3023, 29eqtr3d 2642 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐹𝑞)) = 𝑞)
3130fveq2d 6089 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐺𝑞))
3217, 31eqtr3d 2642 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) = (𝐺𝑞))
3332breq1d 4584 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)𝑥 ↔ (𝐺𝑞)(le‘𝐾)𝑥))
34 simpr1 1059 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑥 ∈ (Base‘𝐾))
35 f1ocnvfv1 6407 . . . . . . . . . . . 12 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3625, 34, 35syl2anc 690 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3736breq2d 4586 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑞)(le‘𝐾)𝑥))
38 f1ocnvfv1 6407 . . . . . . . . . . . 12 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘(𝐺𝑥)) = 𝑥)
397, 34, 38syl2anc 690 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺𝑥)) = 𝑥)
4039breq2d 4586 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥)) ↔ (𝐺𝑞)(le‘𝐾)𝑥))
4133, 37, 403bitr4d 298 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
42 simpl1l 1104 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐾 ∈ HL)
43 eqid 2606 . . . . . . . . . . . 12 (LAut‘𝐾) = (LAut‘𝐾)
444, 43, 5ltrnlaut 34227 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ (LAut‘𝐾))
451, 8, 44syl2anc 690 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹 ∈ (LAut‘𝐾))
463, 4, 5ltrncl 34229 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
471, 8, 34, 46syl3anc 1317 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑥) ∈ (Base‘𝐾))
483, 10, 43lautcnvle 34193 . . . . . . . . . 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 1318 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ (𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥))))
504, 43, 5ltrnlaut 34227 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺 ∈ (LAut‘𝐾))
511, 2, 50syl2anc 690 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺 ∈ (LAut‘𝐾))
523, 4, 5ltrncl 34229 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
531, 2, 34, 52syl3anc 1317 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺𝑥) ∈ (Base‘𝐾))
543, 10, 43lautcnvle 34193 . . . . . . . . . 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 1318 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐺𝑥) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
5641, 49, 553bitr4d 298 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))
57563exp2 1276 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝑥 ∈ (Base‘𝐾) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))))
5857imp 443 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))))
5958ralrimdv 2947 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))
60 simpl1l 1104 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61 simpl1 1056 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
62 simpl2 1057 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹𝑇)
63 simpr 475 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
6461, 62, 63, 46syl3anc 1317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
65 simpl3 1058 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺𝑇)
6661, 65, 63, 52syl3anc 1317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
673, 10, 11hlateq 33503 . . . . . 6 ((𝐾 ∈ HL ∧ (𝐹𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑥) ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6860, 64, 66, 67syl3anc 1317 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6959, 68sylibd 227 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝐹𝑥) = (𝐺𝑥)))
7069ralrimdva 2948 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
71243adant3 1073 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72 f1ofn 6033 . . . . 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 1072 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75 f1ofn 6033 . . . . 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 6201 . . . 4 ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7873, 76, 77syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7970, 78sylibrd 247 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → 𝐹 = 𝐺))
80 fveq1 6084 . . 3 (𝐹 = 𝐺 → (𝐹𝑝) = (𝐺𝑝))
8180ralrimivw 2946 . 2 (𝐹 = 𝐺 → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
8279, 81impbid1 213 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892   class class class wbr 4574  ccnv 5024   Fn wfn 5782  1-1-ontowf1o 5786  cfv 5787  Basecbs 15638  lecple 15718  Atomscatm 33368  HLchlt 33455  LHypclh 34088  LAutclaut 34089  LTrncltrn 34205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-map 7720  df-preset 16694  df-poset 16712  df-plt 16724  df-lub 16740  df-glb 16741  df-join 16742  df-meet 16743  df-p0 16805  df-lat 16812  df-clat 16874  df-oposet 33281  df-ol 33283  df-oml 33284  df-covers 33371  df-ats 33372  df-atl 33403  df-cvlat 33427  df-hlat 33456  df-lhyp 34092  df-laut 34093  df-ldil 34208  df-ltrn 34209
This theorem is referenced by:  ltrneq  34253  cdlemd  34312
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