ltrneq2 - Mathbox for Norm Megill

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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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ↔ 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.

Step	Hyp	Ref	Expression
1		simpl1 1191	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1193	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐺 ∈ 𝑇)
3		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		ltrneq2.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (LHyp‘𝐾)
5		ltrneq2.t	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	3, 4, 5	ltrn1o 38587	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7	1, 2, 6	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8		simpl2 1192	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐹 ∈ 𝑇)
9		simpr3 1196	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝑞 ∈ 𝐴)
10		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
11		ltrneq2.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11, 4, 5	ltrncnvat 38604	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ 𝐴) → (^◡𝐹‘𝑞) ∈ 𝐴)
13	1, 8, 9, 12	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐹‘𝑞) ∈ 𝐴)
14	3, 11	atbase 37751	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘𝑞) ∈ 𝐴 → (^◡𝐹‘𝑞) ∈ (Base‘𝐾))
15	13, 14	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐹‘𝑞) ∈ (Base‘𝐾))
16		f1ocnvfv1 7222	. . . . . . . . . . . . 13 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (^◡𝐹‘𝑞) ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘(^◡𝐹‘𝑞))) = (^◡𝐹‘𝑞))
17	7, 15, 16	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐺‘(𝐺‘(^◡𝐹‘𝑞))) = (^◡𝐹‘𝑞))
18		simpr2 1195	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝))
19		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐹‘𝑝) = (𝐹‘(^◡𝐹‘𝑞)))
20		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐺‘𝑝) = (𝐺‘(^◡𝐹‘𝑞)))
21	19, 20	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (^◡𝐹‘𝑞) → ((𝐹‘𝑝) = (𝐺‘𝑝) ↔ (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞))))
22	21	rspcv 3577	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘𝑞) ∈ 𝐴 → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞))))
23	13, 18, 22	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞)))
24	3, 4, 5	ltrn1o 38587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
25	1, 8, 24	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26	3, 11	atbase 37751	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ 𝐴 → 𝑞 ∈ (Base‘𝐾))
27	9, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝑞 ∈ (Base‘𝐾))
28		f1ocnvfv2 7223	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(^◡𝐹‘𝑞)) = 𝑞)
29	25, 27, 28	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐹‘(^◡𝐹‘𝑞)) = 𝑞)
30	23, 29	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐺‘(^◡𝐹‘𝑞)) = 𝑞)
31	30	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐺‘(𝐺‘(^◡𝐹‘𝑞))) = (^◡𝐺‘𝑞))
32	17, 31	eqtr3d 2778	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐹‘𝑞) = (^◡𝐺‘𝑞))
33	32	breq1d 5115	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → ((^◡𝐹‘𝑞)(le‘𝐾)𝑥 ↔ (^◡𝐺‘𝑞)(le‘𝐾)𝑥))
34		simpr1 1194	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝑥 ∈ (Base‘𝐾))
35		f1ocnvfv1 7222	. . . . . . . . . . . 12 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐹‘(𝐹‘𝑥)) = 𝑥)
36	25, 34, 35	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐹‘(𝐹‘𝑥)) = 𝑥)
37	36	breq2d 5117	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → ((^◡𝐹‘𝑞)(le‘𝐾)(^◡𝐹‘(𝐹‘𝑥)) ↔ (^◡𝐹‘𝑞)(le‘𝐾)𝑥))
38		f1ocnvfv1 7222	. . . . . . . . . . . 12 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
39	7, 34, 38	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
40	39	breq2d 5117	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → ((^◡𝐺‘𝑞)(le‘𝐾)(^◡𝐺‘(𝐺‘𝑥)) ↔ (^◡𝐺‘𝑞)(le‘𝐾)𝑥))
41	33, 37, 40	3bitr4d 310	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → ((^◡𝐹‘𝑞)(le‘𝐾)(^◡𝐹‘(𝐹‘𝑥)) ↔ (^◡𝐺‘𝑞)(le‘𝐾)(^◡𝐺‘(𝐺‘𝑥))))
42		simpl1l 1224	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐾 ∈ HL)
43		eqid 2736	. . . . . . . . . . . 12 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
44	4, 43, 5	ltrnlaut 38586	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
45	1, 8, 44	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐹 ∈ (LAut‘𝐾))
46	3, 4, 5	ltrncl 38588	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
47	1, 8, 34, 46	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
48	3, 10, 43	lautcnvle 38552	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ (𝐹‘𝑥) ∈ (Base‘𝐾))) → (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ (^◡𝐹‘𝑞)(le‘𝐾)(^◡𝐹‘(𝐹‘𝑥))))
49	42, 45, 27, 47, 48	syl22anc 837	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ (^◡𝐹‘𝑞)(le‘𝐾)(^◡𝐹‘(𝐹‘𝑥))))
50	4, 43, 5	ltrnlaut 38586	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ (LAut‘𝐾))
51	1, 2, 50	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐺 ∈ (LAut‘𝐾))
52	3, 4, 5	ltrncl 38588	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
53	1, 2, 34, 52	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
54	3, 10, 43	lautcnvle 38552	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝐺 ∈ (LAut‘𝐾)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ (𝐺‘𝑥) ∈ (Base‘𝐾))) → (𝑞(le‘𝐾)(𝐺‘𝑥) ↔ (^◡𝐺‘𝑞)(le‘𝐾)(^◡𝐺‘(𝐺‘𝑥))))
55	42, 51, 27, 53, 54	syl22anc 837	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝑞(le‘𝐾)(𝐺‘𝑥) ↔ (^◡𝐺‘𝑞)(le‘𝐾)(^◡𝐺‘(𝐺‘𝑥))))
56	41, 49, 55	3bitr4d 310	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥)))
57	56	3exp2 1354	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑥 ∈ (Base‘𝐾) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → (𝑞 ∈ 𝐴 → (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥))))))
58	57	imp 407	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → (𝑞 ∈ 𝐴 → (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥)))))
59	58	ralrimdv 3149	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → ∀𝑞 ∈ 𝐴 (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥))))
60		simpl1l 1224	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61		simpl1 1191	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
62		simpl2 1192	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹 ∈ 𝑇)
63		simpr 485	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
64	61, 62, 63, 46	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
65		simpl3 1193	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺 ∈ 𝑇)
66	61, 65, 63, 52	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
67	3, 10, 11	hlateq 37862	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑥) ∈ (Base‘𝐾) ∧ (𝐺‘𝑥) ∈ (Base‘𝐾)) → (∀𝑞 ∈ 𝐴 (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
68	60, 64, 66, 67	syl3anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞 ∈ 𝐴 (𝑞(le‘𝐾)(𝐹‘𝑥) ↔ 𝑞(le‘𝐾)(𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
69	59, 68	sylibd 238	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → (𝐹‘𝑥) = (𝐺‘𝑥)))
70	69	ralrimdva 3151	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
71	24	3adant3 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72		f1ofn 6785	. . . . 5 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹 Fn (Base‘𝐾))
73	71, 72	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹 Fn (Base‘𝐾))
74	6	3adant2 1131	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75		f1ofn 6785	. . . . 5 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺 Fn (Base‘𝐾))
76	74, 75	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺 Fn (Base‘𝐾))
77		eqfnfv 6982	. . . 4 ⊢ ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
78	73, 76, 77	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
79	70, 78	sylibrd 258	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → 𝐹 = 𝐺))
80		fveq1 6841	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑝) = (𝐺‘𝑝))
81	80	ralrimivw 3147	. 2 ⊢ (𝐹 = 𝐺 → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝))
82	79, 81	impbid1 224	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ↔ 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-onto→wf1o 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

W3C validator