ltrneq2 - Mathbox for Norm Megill

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Theorem ltrneq2 38089

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 1189	. . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐺 ∈ 𝑇)
3		eqid 2738	. . . . . . . . . . . . . . 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 38065	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7	1, 2, 6	syl2anc 583	. . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐹 ∈ 𝑇)
9		simpr3 1194	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝑞 ∈ 𝐴)
10		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
11		ltrneq2.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11, 4, 5	ltrncnvat 38082	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ 𝐴) → (^◡𝐹‘𝑞) ∈ 𝐴)
13	1, 8, 9, 12	syl3anc 1369	. . . . . . . . . . . . . 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 37230	. . . . . . . . . . . . . 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 7129	. . . . . . . . . . . . 13 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (^◡𝐹‘𝑞) ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘(^◡𝐹‘𝑞))) = (^◡𝐹‘𝑞))
17	7, 15, 16	syl2anc 583	. . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐹‘𝑝) = (𝐹‘(^◡𝐹‘𝑞)))
20		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (^◡𝐹‘𝑞) → (𝐺‘𝑝) = (𝐺‘(^◡𝐹‘𝑞)))
21	19, 20	eqeq12d 2754	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (^◡𝐹‘𝑞) → ((𝐹‘𝑝) = (𝐺‘𝑝) ↔ (𝐹‘(^◡𝐹‘𝑞)) = (𝐺‘(^◡𝐹‘𝑞))))
22	21	rspcv 3547	. . . . . . . . . . . . . . 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 38065	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
25	1, 8, 24	syl2anc 583	. . . . . . . . . . . . . . 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 37230	. . . . . . . . . . . . . . . 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 7130	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(^◡𝐹‘𝑞)) = 𝑞)
29	25, 27, 28	syl2anc 583	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 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 6760	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 5080	. . . . . . . . . 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 1192	. . . . . . . . . . . 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 7129	. . . . . . . . . . . 12 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐹‘(𝐹‘𝑥)) = 𝑥)
36	25, 34, 35	syl2anc 583	. . . . . . . . . . 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 5082	. . . . . . . . . 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 7129	. . . . . . . . . . . 12 ⊢ ((𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
39	7, 34, 38	syl2anc 583	. . . . . . . . . . 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 5082	. . . . . . . . . 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 1222	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ∧ 𝑞 ∈ 𝐴)) → 𝐾 ∈ HL)
43		eqid 2738	. . . . . . . . . . . 12 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
44	4, 43, 5	ltrnlaut 38064	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
45	1, 8, 44	syl2anc 583	. . . . . . . . . 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 38066	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
47	1, 8, 34, 46	syl3anc 1369	. . . . . . . . . 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 38030	. . . . . . . . . 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 835	. . . . . . . . 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 38064	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ (LAut‘𝐾))
51	1, 2, 50	syl2anc 583	. . . . . . . . . 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 38066	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
53	1, 2, 34, 52	syl3anc 1369	. . . . . . . . . 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 38030	. . . . . . . . . 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 835	. . . . . . . . 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 1352	. . . . . . 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 406	. . . . . 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 3111	. . . . 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 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ HL)
61		simpl1 1189	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
62		simpl2 1190	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐹 ∈ 𝑇)
63		simpr 484	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
64	61, 62, 63, 46	syl3anc 1369	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐹‘𝑥) ∈ (Base‘𝐾))
65		simpl3 1191	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐺 ∈ 𝑇)
66	61, 65, 63, 52	syl3anc 1369	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝐺‘𝑥) ∈ (Base‘𝐾))
67	3, 10, 11	hlateq 37340	. . . . . 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 1369	. . . . 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 3112	. . 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 1130	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
72		f1ofn 6701	. . . . 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 1129	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
75		f1ofn 6701	. . . . 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 6891	. . . 4 ⊢ ((𝐹 Fn (Base‘𝐾) ∧ 𝐺 Fn (Base‘𝐾)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (Base‘𝐾)(𝐹‘𝑥) = (𝐺‘𝑥)))
78	73, 76, 77	syl2anc 583	. . 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 6755	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑝) = (𝐺‘𝑝))
81	80	ralrimivw 3108	. 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 class class class wbr 5070 ^◡ccnv 5579 Fn wfn 6413 –1-1-onto→wf1o 6417 ‘cfv 6418 Basecbs 16840 lecple 16895 Atomscatm 37204 HLchlt 37291 LHypclh 37925 LAutclaut 37926 LTrncltrn 38042

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-lat 18065 df-clat 18132 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046

This theorem is referenced by: ltrneq 38090 cdlemd 38148

W3C validator