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Theorem iscgrglt 28390

Description: The property for two sequences 𝐴 and 𝐵 of points to be congruent, where the congruence is only required for indices verifying a less-than relation. (Contributed by Thierry Arnoux, 7-Oct-2020.)

**Hypotheses**
Ref	Expression
trgcgrg.p	⊢ 𝑃 = (Base‘𝐺)
trgcgrg.m	⊢ − = (dist‘𝐺)
trgcgrg.r	⊢ ∼ = (cgrG‘𝐺)
trgcgrg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
iscgrglt.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
iscgrglt.a	⊢ (𝜑 → 𝐴:𝐷⟶𝑃)
iscgrglt.b	⊢ (𝜑 → 𝐵:𝐷⟶𝑃)

**Assertion**
Ref	Expression
iscgrglt	⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))

Distinct variable groups: − ,𝑖,𝑗 𝐴,𝑖,𝑗 𝐵,𝑖,𝑗 𝑖,𝐺,𝑗 𝜑,𝑖,𝑗 Allowed substitution hints: 𝐷(𝑖,𝑗) 𝑃(𝑖,𝑗) ∼ (𝑖,𝑗)

Proof of Theorem iscgrglt Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trgcgrg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		trgcgrg.m	. . 3 ⊢ − = (dist‘𝐺)
3		trgcgrg.r	. . 3 ⊢ ∼ = (cgrG‘𝐺)
4		trgcgrg.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		iscgrglt.d	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
6		iscgrglt.a	. . 3 ⊢ (𝜑 → 𝐴:𝐷⟶𝑃)
7		iscgrglt.b	. . 3 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃)
8	1, 2, 3, 4, 5, 6, 7	iscgrgd 28389	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
9		simp2 1134	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ dom 𝐴 ∧ 𝑗 ∈ dom 𝐴)) ∧ ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) ∧ 𝑖 < 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
10	9	3exp 1116	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐴 ∧ 𝑗 ∈ dom 𝐴)) → (((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) → (𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
11	10	ralimdvva 3194	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) → ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
12		breq1 5152	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘 < 𝑙 ↔ 𝑖 < 𝑙))
13		fveq2 6896	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝐴‘𝑘) = (𝐴‘𝑖))
14	13	oveq1d 7434	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐴‘𝑖) − (𝐴‘𝑙)))
15		fveq2 6896	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝐵‘𝑘) = (𝐵‘𝑖))
16	15	oveq1d 7434	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((𝐵‘𝑘) − (𝐵‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙)))
17	14, 16	eqeq12d 2741	. . . . . 6 ⊢ (𝑘 = 𝑖 → (((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙))))
18	12, 17	imbi12d 343	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))) ↔ (𝑖 < 𝑙 → ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙)))))
19		breq2 5153	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝑖 < 𝑙 ↔ 𝑖 < 𝑗))
20		fveq2 6896	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝐴‘𝑙) = (𝐴‘𝑗))
21	20	oveq2d 7435	. . . . . . 7 ⊢ (𝑙 = 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐴‘𝑖) − (𝐴‘𝑗)))
22		fveq2 6896	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝐵‘𝑙) = (𝐵‘𝑗))
23	22	oveq2d 7435	. . . . . . 7 ⊢ (𝑙 = 𝑗 → ((𝐵‘𝑖) − (𝐵‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
24	21, 23	eqeq12d 2741	. . . . . 6 ⊢ (𝑙 = 𝑗 → (((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
25	19, 24	imbi12d 343	. . . . 5 ⊢ (𝑙 = 𝑗 → ((𝑖 < 𝑙 → ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙))) ↔ (𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
26	18, 25	cbvral2vw 3228	. . . 4 ⊢ (∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))) ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
27		simpllr 774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 ∈ dom 𝐴)
28		simplr 767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑗 ∈ dom 𝐴)
29		simp-4r 782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))))
30	27, 28, 29	jca31 513	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ((𝑖 ∈ dom 𝐴 ∧ 𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))))
31		simpr 483	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
32	18, 25	rspc2va 3618	. . . . . . . . 9 ⊢ (((𝑖 ∈ dom 𝐴 ∧ 𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) → (𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
33	30, 31, 32	sylc 65	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
34		eqid 2725	. . . . . . . . . . 11 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
35	4	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝐺 ∈ TarskiG)
36	6	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐴:𝐷⟶𝑃)
37		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
38	36	fdmd 6733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 = 𝐷)
39	37, 38	eleqtrd 2827	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ 𝐷)
40	36, 39	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑖) ∈ 𝑃)
41	40	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴‘𝑖) ∈ 𝑃)
42	7	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐵:𝐷⟶𝑃)
43	42, 39	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵‘𝑖) ∈ 𝑃)
44	43	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵‘𝑖) ∈ 𝑃)
45	1, 2, 34, 35, 41, 44	tgcgrtriv 28360	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑖)) = ((𝐵‘𝑖) − (𝐵‘𝑖)))
46		simpr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
47	46	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴‘𝑖) = (𝐴‘𝑗))
48	47	oveq2d 7435	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑖)) = ((𝐴‘𝑖) − (𝐴‘𝑗)))
49	46	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵‘𝑖) = (𝐵‘𝑗))
50	49	oveq2d 7435	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐵‘𝑖) − (𝐵‘𝑖)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
51	45, 48, 50	3eqtr3d 2773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
52	51	adantl3r 748	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
53	4	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝐺 ∈ TarskiG)
54		simpr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
55	54, 38	eleqtrd 2827	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ 𝐷)
56	36, 55	ffvelcdmd 7094	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑗) ∈ 𝑃)
57	56	adantr 479	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑗) ∈ 𝑃)
58	57	adantl3r 748	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑗) ∈ 𝑃)
59	40	adantr 479	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑖) ∈ 𝑃)
60	59	adantl3r 748	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑖) ∈ 𝑃)
61	42, 55	ffvelcdmd 7094	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵‘𝑗) ∈ 𝑃)
62	61	adantr 479	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑗) ∈ 𝑃)
63	62	adantl3r 748	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑗) ∈ 𝑃)
64	43	adantr 479	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑖) ∈ 𝑃)
65	64	adantl3r 748	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑖) ∈ 𝑃)
66		simplr 767	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 ∈ dom 𝐴)
67		simpllr 774	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑖 ∈ dom 𝐴)
68		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))))
69	66, 67, 68	jca31 513	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝑗 ∈ dom 𝐴 ∧ 𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))))
70		simpr 483	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
71		breq1 5152	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑘 < 𝑙 ↔ 𝑗 < 𝑙))
72		fveq2 6896	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
73	72	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐴‘𝑗) − (𝐴‘𝑙)))
74		fveq2 6896	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
75	74	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝐵‘𝑘) − (𝐵‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙)))
76	73, 75	eqeq12d 2741	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙))))
77	71, 76	imbi12d 343	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))) ↔ (𝑗 < 𝑙 → ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙)))))
78		breq2 5153	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑗 < 𝑙 ↔ 𝑗 < 𝑖))
79		fveq2 6896	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑖 → (𝐴‘𝑙) = (𝐴‘𝑖))
80	79	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐴‘𝑗) − (𝐴‘𝑖)))
81		fveq2 6896	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑖 → (𝐵‘𝑙) = (𝐵‘𝑖))
82	81	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → ((𝐵‘𝑗) − (𝐵‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑖)))
83	80, 82	eqeq12d 2741	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑗) − (𝐴‘𝑖)) = ((𝐵‘𝑗) − (𝐵‘𝑖))))
84	78, 83	imbi12d 343	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → ((𝑗 < 𝑙 → ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙))) ↔ (𝑗 < 𝑖 → ((𝐴‘𝑗) − (𝐴‘𝑖)) = ((𝐵‘𝑗) − (𝐵‘𝑖)))))
85	77, 84	rspc2va 3618	. . . . . . . . . 10 ⊢ (((𝑗 ∈ dom 𝐴 ∧ 𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) → (𝑗 < 𝑖 → ((𝐴‘𝑗) − (𝐴‘𝑖)) = ((𝐵‘𝑗) − (𝐵‘𝑖))))
86	69, 70, 85	sylc 65	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝐴‘𝑗) − (𝐴‘𝑖)) = ((𝐵‘𝑗) − (𝐵‘𝑖)))
87	1, 2, 34, 53, 58, 60, 63, 65, 86	tgcgrcomlr 28356	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
88	6	fdmd 6733	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐴 = 𝐷)
89	88, 5	eqsstrd 4015	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐴 ⊆ ℝ)
90	89	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 ⊆ ℝ)
91		simplr 767	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
92	90, 91	sseldd 3977	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ ℝ)
93		simpr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
94	90, 93	sseldd 3977	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ ℝ)
95	92, 94	lttri4d 11387	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖))
96	33, 52, 87, 95	mpjao3dan 1428	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
97	96	anasss 465	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) ∧ (𝑖 ∈ dom 𝐴 ∧ 𝑗 ∈ dom 𝐴)) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
98	97	ralrimivva 3190	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)))) → ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
99	98	ex 411	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ dom 𝐴∀𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙))) → ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
100	26, 99	biimtrrid 242	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))) → ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
101	11, 100	impbid 211	. 2 ⊢ (𝜑 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
102	8, 101	bitrd 278	1 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 class class class wbr 5149 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 < clt 11280 Basecbs 17183 distcds 17245 TarskiGcstrkg 28303 Itvcitv 28309 cgrGccgrg 28386

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-trkgc 28324 df-trkgcb 28326 df-trkg 28329 df-cgrg 28387

This theorem is referenced by: tgcgr4 28407

W3C validator