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Theorem iscgrglt 26286
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 (𝜑𝐺 ∈ TarskiG)
iscgrglt.d (𝜑𝐷 ⊆ ℝ)
iscgrglt.a (𝜑𝐴:𝐷𝑃)
iscgrglt.b (𝜑𝐵:𝐷𝑃)
Assertion
Ref Expression
iscgrglt (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
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.
StepHypRef Expression
1 trgcgrg.p . . 3 𝑃 = (Base‘𝐺)
2 trgcgrg.m . . 3 = (dist‘𝐺)
3 trgcgrg.r . . 3 = (cgrG‘𝐺)
4 trgcgrg.g . . 3 (𝜑𝐺 ∈ TarskiG)
5 iscgrglt.d . . 3 (𝜑𝐷 ⊆ ℝ)
6 iscgrglt.a . . 3 (𝜑𝐴:𝐷𝑃)
7 iscgrglt.b . . 3 (𝜑𝐵:𝐷𝑃)
81, 2, 3, 4, 5, 6, 7iscgrgd 26285 . 2 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
9 simp2 1134 . . . . 5 (((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) ∧ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ∧ 𝑖 < 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
1093exp 1116 . . . 4 ((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → (((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1110ralimdvva 3167 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
12 breq1 5042 . . . . . 6 (𝑘 = 𝑖 → (𝑘 < 𝑙𝑖 < 𝑙))
13 fveq2 6643 . . . . . . . 8 (𝑘 = 𝑖 → (𝐴𝑘) = (𝐴𝑖))
1413oveq1d 7145 . . . . . . 7 (𝑘 = 𝑖 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑙)))
15 fveq2 6643 . . . . . . . 8 (𝑘 = 𝑖 → (𝐵𝑘) = (𝐵𝑖))
1615oveq1d 7145 . . . . . . 7 (𝑘 = 𝑖 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑙)))
1714, 16eqeq12d 2837 . . . . . 6 (𝑘 = 𝑖 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))))
1812, 17imbi12d 348 . . . . 5 (𝑘 = 𝑖 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)))))
19 breq2 5043 . . . . . 6 (𝑙 = 𝑗 → (𝑖 < 𝑙𝑖 < 𝑗))
20 fveq2 6643 . . . . . . . 8 (𝑙 = 𝑗 → (𝐴𝑙) = (𝐴𝑗))
2120oveq2d 7146 . . . . . . 7 (𝑙 = 𝑗 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑗)))
22 fveq2 6643 . . . . . . . 8 (𝑙 = 𝑗 → (𝐵𝑙) = (𝐵𝑗))
2322oveq2d 7146 . . . . . . 7 (𝑙 = 𝑗 → ((𝐵𝑖) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑗)))
2421, 23eqeq12d 2837 . . . . . 6 (𝑙 = 𝑗 → (((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
2519, 24imbi12d 348 . . . . 5 (𝑙 = 𝑗 → ((𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))) ↔ (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
2618, 25cbvral2vw 3438 . . . 4 (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
27 simpllr 775 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 ∈ dom 𝐴)
28 simplr 768 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑗 ∈ dom 𝐴)
29 simp-4r 783 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))))
3027, 28, 29jca31 518 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))))
31 simpr 488 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
3218, 25rspc2va 3611 . . . . . . . . 9 (((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
3330, 31, 32sylc 65 . . . . . . . 8 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
34 eqid 2821 . . . . . . . . . . 11 (Itv‘𝐺) = (Itv‘𝐺)
354ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝐺 ∈ TarskiG)
366ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐴:𝐷𝑃)
37 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
3836fdmd 6496 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 = 𝐷)
3937, 38eleqtrd 2914 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖𝐷)
4036, 39ffvelrnd 6825 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑖) ∈ 𝑃)
4140adantr 484 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) ∈ 𝑃)
427ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐵:𝐷𝑃)
4342, 39ffvelrnd 6825 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑖) ∈ 𝑃)
4443adantr 484 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) ∈ 𝑃)
451, 2, 34, 35, 41, 44tgcgrtriv 26256 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐵𝑖) (𝐵𝑖)))
46 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
4746fveq2d 6647 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) = (𝐴𝑗))
4847oveq2d 7146 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐴𝑖) (𝐴𝑗)))
4946fveq2d 6647 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) = (𝐵𝑗))
5049oveq2d 7146 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐵𝑖) (𝐵𝑖)) = ((𝐵𝑖) (𝐵𝑗)))
5145, 48, 503eqtr3d 2864 . . . . . . . . 9 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
5251adantl3r 749 . . . . . . . 8 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
534ad4antr 731 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝐺 ∈ TarskiG)
54 simpr 488 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
5554, 38eleqtrd 2914 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗𝐷)
5636, 55ffvelrnd 6825 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑗) ∈ 𝑃)
5756adantr 484 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
5857adantl3r 749 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
5940adantr 484 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6059adantl3r 749 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6142, 55ffvelrnd 6825 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑗) ∈ 𝑃)
6261adantr 484 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6362adantl3r 749 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6443adantr 484 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
6564adantl3r 749 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
66 simplr 768 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 ∈ dom 𝐴)
67 simpllr 775 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑖 ∈ dom 𝐴)
68 simp-4r 783 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))))
6966, 67, 68jca31 518 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))))
70 simpr 488 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
71 breq1 5042 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑘 < 𝑙𝑗 < 𝑙))
72 fveq2 6643 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
7372oveq1d 7145 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑙)))
74 fveq2 6643 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝐵𝑘) = (𝐵𝑗))
7574oveq1d 7145 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑙)))
7673, 75eqeq12d 2837 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))))
7771, 76imbi12d 348 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)))))
78 breq2 5043 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑗 < 𝑙𝑗 < 𝑖))
79 fveq2 6643 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (𝐴𝑙) = (𝐴𝑖))
8079oveq2d 7146 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑖)))
81 fveq2 6643 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (𝐵𝑙) = (𝐵𝑖))
8281oveq2d 7146 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → ((𝐵𝑗) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑖)))
8380, 82eqeq12d 2837 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
8478, 83imbi12d 348 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))) ↔ (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))))
8577, 84rspc2va 3611 . . . . . . . . . 10 (((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
8669, 70, 85sylc 65 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))
871, 2, 34, 53, 58, 60, 63, 65, 86tgcgrcomlr 26252 . . . . . . . 8 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
886fdmd 6496 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝐷)
8988, 5eqsstrd 3981 . . . . . . . . . . 11 (𝜑 → dom 𝐴 ⊆ ℝ)
9089ad3antrrr 729 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 ⊆ ℝ)
91 simplr 768 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
9290, 91sseldd 3944 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ ℝ)
93 simpr 488 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
9490, 93sseldd 3944 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ ℝ)
9592, 94lttri4d 10758 . . . . . . . 8 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
9633, 52, 87, 95mpjao3dan 1428 . . . . . . 7 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
9796anasss 470 . . . . . 6 (((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
9897ralrimivva 3179 . . . . 5 ((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
9998ex 416 . . . 4 (𝜑 → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10026, 99syl5bir 246 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10111, 100impbid 215 . 2 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1028, 101bitrd 282 1 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wss 3910   class class class wbr 5039  dom cdm 5528  wf 6324  cfv 6328  (class class class)co 7130  cr 10513   < clt 10652  Basecbs 16461  distcds 16552  TarskiGcstrkg 26202  Itvcitv 26208  cgrGccgrg 26282
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-pre-lttri 10588  ax-pre-lttrn 10589
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-er 8264  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-ltxr 10657  df-trkgc 26220  df-trkgcb 26222  df-trkg 26225  df-cgrg 26283
This theorem is referenced by:  tgcgr4  26303
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