ltrneq2 - Mathbox for Norm Megill

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Theorem ltrneq2 34452

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 1057	. . . . . . . . . . . . . 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 1059	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐺 ∈ 𝑇)
3		eqid 2610	. . . . . . . . . . . . . . 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 34428	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7	1, 2, 6	syl2anc 691	. . . . . . . . . . . . 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 1058	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐹 ∈ 𝑇)
9		simpr3 1062	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝑞 ∈ 𝐴)
10		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
11		ltrneq2.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11, 4, 5	ltrncnvat 34445	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ 𝐴) → (^◡𝐹‘𝑞) ∈ 𝐴)
13	1, 8, 9, 12	syl3anc 1318	. . . . . . . . . . . . . 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 33594	. . . . . . . . . . . . . 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 6432	. . . . . . . . . . . . 13 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (^◡𝐹‘𝑞) ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘(^◡𝐹‘𝑞))) = (^◡𝐹‘𝑞))
17	7, 15, 16	syl2anc 691	. . . . . . . . . . . 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 1061	. . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐹‘𝑝) = (𝐹‘(^◡𝐹‘𝑞)))
20		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐺‘𝑝) = (𝐺‘(^◡𝐹‘𝑞)))
21	19, 20	eqeq12d 2625	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (^◡𝐹‘𝑞) → ((𝐹‘𝑝) = (𝐺‘𝑝) ↔ (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞))))
22	21	rspcv 3278	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘𝑞) ∈ 𝐴 → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞))))
23	13, 18, 22	sylc 63	. . . . . . . . . . . . . 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 34428	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
25	1, 8, 24	syl2anc 691	. . . . . . . . . . . . . . 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 33594	. . . . . . . . . . . . . . . 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 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(^◡𝐹‘𝑞)) = 𝑞)
29	25, 27, 28	syl2anc 691	. . . . . . . . . . . . . 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 2646	. . . . . . . . . . . . 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 6107	. . . . . . . . . . . 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 2646	. . . . . . . . . . 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 4593	. . . . . . . . . 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 1060	. . . . . . . . . . . 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 6432	. . . . . . . . . . . 12 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐹‘(𝐹‘𝑥)) = 𝑥)
36	25, 34, 35	syl2anc 691	. . . . . . . . . . 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 4595	. . . . . . . . . 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 6432	. . . . . . . . . . . 12 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
39	7, 34, 38	syl2anc 691	. . . . . . . . . . 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 4595	. . . . . . . . . 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 299	. . . . . . . . 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 1105	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐾 ∈ HL)
43		eqid 2610	. . . . . . . . . . . 12 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
44	4, 43, 5	ltrnlaut 34427	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
45	1, 8, 44	syl2anc 691	. . . . . . . . . 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 34429	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
47	1, 8, 34, 46	syl3anc 1318	. . . . . . . . . 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 34393	. . . . . . . . . 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 1319	. . . . . . . . 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 34427	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ (LAut‘𝐾))
51	1, 2, 50	syl2anc 691	. . . . . . . . . 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 34429	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
53	1, 2, 34, 52	syl3anc 1318	. . . . . . . . . 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 34393	. . . . . . . . . 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 1319	. . . . . . . . 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 299	. . . . . . . 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 1277	. . . . . . 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 444	. . . . . 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 2951	. . . . 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 1105	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61		simpl1 1057	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
62		simpl2 1058	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹 ∈ 𝑇)
63		simpr 476	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
64	61, 62, 63, 46	syl3anc 1318	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
65		simpl3 1059	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺 ∈ 𝑇)
66	61, 65, 63, 52	syl3anc 1318	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
67	3, 10, 11	hlateq 33703	. . . . . 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 1318	. . . . 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 228	. . . 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 2952	. . 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 1074	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72		f1ofn 6051	. . . . 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 1073	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75		f1ofn 6051	. . . . 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 6219	. . . 4 ⊢ ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
78	73, 76, 77	syl2anc 691	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
79	70, 78	sylibrd 248	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) → 𝐹 = 𝐺))
80		fveq1 6102	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑝) = (𝐺‘𝑝))
81	80	ralrimivw 2950	. 2 ⊢ (𝐹 = 𝐺 → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝))
82	79, 81	impbid1 214	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ^◡ccnv 5037 Fn wfn 5799 –1-1-onto→wf1o 5803 ‘cfv 5804 Basecbs 15695 lecple 15775 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LAutclaut 34289 LTrncltrn 34405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409

This theorem is referenced by: ltrneq 34453 cdlemd 34512

W3C validator