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Theorem iscgrglt 28492

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 28491	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
9		simp2 1137	. . . . 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 1119	. . . 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 3179	. . 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 5092	. . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘 < 𝑙 ↔ 𝑖 < 𝑙))
13		fveq2 6822	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝐴‘𝑘) = (𝐴‘𝑖))
14	13	oveq1d 7361	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐴‘𝑖) − (𝐴‘𝑙)))
15		fveq2 6822	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝐵‘𝑘) = (𝐵‘𝑖))
16	15	oveq1d 7361	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((𝐵‘𝑘) − (𝐵‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙)))
17	14, 16	eqeq12d 2747	. . . . . 6 ⊢ (𝑘 = 𝑖 → (((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙))))
18	12, 17	imbi12d 344	. . . . 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 5093	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝑖 < 𝑙 ↔ 𝑖 < 𝑗))
20		fveq2 6822	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝐴‘𝑙) = (𝐴‘𝑗))
21	20	oveq2d 7362	. . . . . . 7 ⊢ (𝑙 = 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐴‘𝑖) − (𝐴‘𝑗)))
22		fveq2 6822	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝐵‘𝑙) = (𝐵‘𝑗))
23	22	oveq2d 7362	. . . . . . 7 ⊢ (𝑙 = 𝑗 → ((𝐵‘𝑖) − (𝐵‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
24	21, 23	eqeq12d 2747	. . . . . 6 ⊢ (𝑙 = 𝑗 → (((𝐴‘𝑖) − (𝐴‘𝑙)) = ((𝐵‘𝑖) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
25	19, 24	imbi12d 344	. . . . 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 3214	. . . 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 775	. . . . . . . . . 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 768	. . . . . . . . . 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 783	. . . . . . . . . 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 514	. . . . . . . . 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 484	. . . . . . . . 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 3584	. . . . . . . . 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 2731	. . . . . . . . . . 11 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
35	4	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝐺 ∈ TarskiG)
36	6	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐴:𝐷⟶𝑃)
37		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
38	36	fdmd 6661	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 = 𝐷)
39	37, 38	eleqtrd 2833	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ 𝐷)
40	36, 39	ffvelcdmd 7018	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑖) ∈ 𝑃)
41	40	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴‘𝑖) ∈ 𝑃)
42	7	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐵:𝐷⟶𝑃)
43	42, 39	ffvelcdmd 7018	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵‘𝑖) ∈ 𝑃)
44	43	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵‘𝑖) ∈ 𝑃)
45	1, 2, 34, 35, 41, 44	tgcgrtriv 28462	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑖)) = ((𝐵‘𝑖) − (𝐵‘𝑖)))
46		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
47	46	fveq2d 6826	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴‘𝑖) = (𝐴‘𝑗))
48	47	oveq2d 7362	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑖)) = ((𝐴‘𝑖) − (𝐴‘𝑗)))
49	46	fveq2d 6826	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵‘𝑖) = (𝐵‘𝑗))
50	49	oveq2d 7362	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐵‘𝑖) − (𝐵‘𝑖)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
51	45, 48, 50	3eqtr3d 2774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
52	51	adantl3r 750	. . . . . . . 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 732	. . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
55	54, 38	eleqtrd 2833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ 𝐷)
56	36, 55	ffvelcdmd 7018	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑗) ∈ 𝑃)
57	56	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑗) ∈ 𝑃)
58	57	adantl3r 750	. . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴‘𝑖) ∈ 𝑃)
60	59	adantl3r 750	. . . . . . . . 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 7018	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵‘𝑗) ∈ 𝑃)
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑗) ∈ 𝑃)
63	62	adantl3r 750	. . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵‘𝑖) ∈ 𝑃)
65	64	adantl3r 750	. . . . . . . . 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 768	. . . . . . . . . . 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 775	. . . . . . . . . . 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 783	. . . . . . . . . . 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 514	. . . . . . . . . 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 484	. . . . . . . . . 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 5092	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑘 < 𝑙 ↔ 𝑗 < 𝑙))
72		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
73	72	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐴‘𝑗) − (𝐴‘𝑙)))
74		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
75	74	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝐵‘𝑘) − (𝐵‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙)))
76	73, 75	eqeq12d 2747	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (((𝐴‘𝑘) − (𝐴‘𝑙)) = ((𝐵‘𝑘) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙))))
77	71, 76	imbi12d 344	. . . . . . . . . . 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 5093	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑗 < 𝑙 ↔ 𝑗 < 𝑖))
79		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑖 → (𝐴‘𝑙) = (𝐴‘𝑖))
80	79	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → ((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐴‘𝑗) − (𝐴‘𝑖)))
81		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑖 → (𝐵‘𝑙) = (𝐵‘𝑖))
82	81	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → ((𝐵‘𝑗) − (𝐵‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑖)))
83	80, 82	eqeq12d 2747	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (((𝐴‘𝑗) − (𝐴‘𝑙)) = ((𝐵‘𝑗) − (𝐵‘𝑙)) ↔ ((𝐴‘𝑗) − (𝐴‘𝑖)) = ((𝐵‘𝑗) − (𝐵‘𝑖))))
84	78, 83	imbi12d 344	. . . . . . . . . . 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 3584	. . . . . . . . . 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 28458	. . . . . . . 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 6661	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐴 = 𝐷)
89	88, 5	eqsstrd 3964	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐴 ⊆ ℝ)
90	89	ad3antrrr 730	. . . . . . . . . 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 768	. . . . . . . . . 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 3930	. . . . . . . . 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 484	. . . . . . . . . 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 3930	. . . . . . . . 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 11254	. . . . . . . 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 1434	. . . . . . 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 466	. . . . . 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 3175	. . . . 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 412	. . . 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 243	. . 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 212	. 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 279	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ⊆ wss 3897 class class class wbr 5089 dom cdm 5614 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 < clt 11146 Basecbs 17120 distcds 17170 TarskiGcstrkg 28405 Itvcitv 28411 cgrGccgrg 28488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-er 8622 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-ltxr 11151 df-trkgc 28426 df-trkgcb 28428 df-trkg 28431 df-cgrg 28489

This theorem is referenced by: tgcgr4 28509

W3C validator