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Theorem ltrneq2 36036
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 1242 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simpl3 1246 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺𝑇)
3 eqid 2764 . . . . . . . . . . . . . . 15 (Base‘𝐾) = (Base‘𝐾)
4 ltrneq2.h . . . . . . . . . . . . . . 15 𝐻 = (LHyp‘𝐾)
5 ltrneq2.t . . . . . . . . . . . . . . 15 𝑇 = ((LTrn‘𝐾)‘𝑊)
63, 4, 5ltrn1o 36012 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
71, 2, 6syl2anc 579 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8 simpl2 1244 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹𝑇)
9 simpr3 1252 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞𝐴)
10 eqid 2764 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
11 ltrneq2.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
1210, 11, 4, 5ltrncnvat 36029 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞𝐴) → (𝐹𝑞) ∈ 𝐴)
131, 8, 9, 12syl3anc 1490 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) ∈ 𝐴)
143, 11atbase 35177 . . . . . . . . . . . . . 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 6723 . . . . . . . . . . . . 13 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (𝐹𝑞) ∈ (Base‘𝐾)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
177, 15, 16syl2anc 579 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐹𝑞))
18 simpr2 1250 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
19 fveq2 6374 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐹𝑝) = (𝐹‘(𝐹𝑞)))
20 fveq2 6374 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝐹𝑞) → (𝐺𝑝) = (𝐺‘(𝐹𝑞)))
2119, 20eqeq12d 2779 . . . . . . . . . . . . . . . 16 (𝑝 = (𝐹𝑞) → ((𝐹𝑝) = (𝐺𝑝) ↔ (𝐹‘(𝐹𝑞)) = (𝐺‘(𝐹𝑞))))
2221rspcv 3456 . . . . . . . . . . . . . . 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 36012 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
251, 8, 24syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
263, 11atbase 35177 . . . . . . . . . . . . . . . 16 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
279, 26syl 17 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
28 f1ocnvfv2 6724 . . . . . . . . . . . . . . 15 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
2925, 27, 28syl2anc 579 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑞)) = 𝑞)
3023, 29eqtr3d 2800 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐹𝑞)) = 𝑞)
3130fveq2d 6378 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺‘(𝐹𝑞))) = (𝐺𝑞))
3217, 31eqtr3d 2800 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑞) = (𝐺𝑞))
3332breq1d 4818 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)𝑥 ↔ (𝐺𝑞)(le‘𝐾)𝑥))
34 simpr1 1248 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝑥 ∈ (Base‘𝐾))
35 f1ocnvfv1 6723 . . . . . . . . . . . 12 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3625, 34, 35syl2anc 579 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹‘(𝐹𝑥)) = 𝑥)
3736breq2d 4820 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐹𝑞)(le‘𝐾)𝑥))
38 f1ocnvfv1 6723 . . . . . . . . . . . 12 ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘(𝐺𝑥)) = 𝑥)
397, 34, 38syl2anc 579 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺‘(𝐺𝑥)) = 𝑥)
4039breq2d 4820 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥)) ↔ (𝐺𝑞)(le‘𝐾)𝑥))
4133, 37, 403bitr4d 302 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → ((𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥)) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
42 simpl1l 1293 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐾 ∈ HL)
43 eqid 2764 . . . . . . . . . . . 12 (LAut‘𝐾) = (LAut‘𝐾)
444, 43, 5ltrnlaut 36011 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ (LAut‘𝐾))
451, 8, 44syl2anc 579 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐹 ∈ (LAut‘𝐾))
463, 4, 5ltrncl 36013 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
471, 8, 34, 46syl3anc 1490 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐹𝑥) ∈ (Base‘𝐾))
483, 10, 43lautcnvle 35977 . . . . . . . . . 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 867 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ (𝐹𝑞)(le‘𝐾)(𝐹‘(𝐹𝑥))))
504, 43, 5ltrnlaut 36011 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺 ∈ (LAut‘𝐾))
511, 2, 50syl2anc 579 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → 𝐺 ∈ (LAut‘𝐾))
523, 4, 5ltrncl 36013 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
531, 2, 34, 52syl3anc 1490 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝐺𝑥) ∈ (Base‘𝐾))
543, 10, 43lautcnvle 35977 . . . . . . . . . 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 867 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐺𝑥) ↔ (𝐺𝑞)(le‘𝐾)(𝐺‘(𝐺𝑥))))
5641, 49, 553bitr4d 302 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ∧ 𝑞𝐴)) → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))
57563exp2 1463 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝑥 ∈ (Base‘𝐾) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))))
5857imp 395 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝑞𝐴 → (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)))))
5958ralrimdv 3114 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥))))
60 simpl1l 1293 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61 simpl1 1242 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
62 simpl2 1244 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹𝑇)
63 simpr 477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
6461, 62, 63, 46syl3anc 1490 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹𝑥) ∈ (Base‘𝐾))
65 simpl3 1246 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺𝑇)
6661, 65, 63, 52syl3anc 1490 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺𝑥) ∈ (Base‘𝐾))
673, 10, 11hlateq 35287 . . . . . 6 ((𝐾 ∈ HL ∧ (𝐹𝑥) ∈ (Base‘𝐾) ∧ (𝐺𝑥) ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6860, 64, 66, 67syl3anc 1490 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞𝐴 (𝑞(le‘𝐾)(𝐹𝑥) ↔ 𝑞(le‘𝐾)(𝐺𝑥)) ↔ (𝐹𝑥) = (𝐺𝑥)))
6959, 68sylibd 230 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → (𝐹𝑥) = (𝐺𝑥)))
7069ralrimdva 3115 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
71243adant3 1162 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72 f1ofn 6320 . . . . 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 1161 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75 f1ofn 6320 . . . . 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 6500 . . . 4 ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7873, 76, 77syl2anc 579 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹𝑥) = (𝐺𝑥)))
7970, 78sylibrd 250 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) → 𝐹 = 𝐺))
80 fveq1 6373 . . 3 (𝐹 = 𝐺 → (𝐹𝑝) = (𝐺𝑝))
8180ralrimivw 3113 . 2 (𝐹 = 𝐺 → ∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝))
8279, 81impbid1 216 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3054   class class class wbr 4808  ccnv 5275   Fn wfn 6062  1-1-ontowf1o 6066  cfv 6067  Basecbs 16131  lecple 16222  Atomscatm 35151  HLchlt 35238  LHypclh 35872  LAutclaut 35873  LTrncltrn 35989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-map 8061  df-proset 17195  df-poset 17213  df-plt 17225  df-lub 17241  df-glb 17242  df-join 17243  df-meet 17244  df-p0 17306  df-lat 17313  df-clat 17375  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-lhyp 35876  df-laut 35877  df-ldil 35992  df-ltrn 35993
This theorem is referenced by:  ltrneq  36037  cdlemd  36095
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