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Theorem iscgrglt 25700
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 25699 . 2 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
9 simp2 1167 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) ∧ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ∧ 𝑖 < 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
1093expia 1150 . . . . 5 (((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) ∧ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
1110ex 401 . . . 4 ((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → (((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1211ralimdvva 3111 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
13 breq1 4812 . . . . . 6 (𝑘 = 𝑖 → (𝑘 < 𝑙𝑖 < 𝑙))
14 fveq2 6375 . . . . . . . 8 (𝑘 = 𝑖 → (𝐴𝑘) = (𝐴𝑖))
1514oveq1d 6857 . . . . . . 7 (𝑘 = 𝑖 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑙)))
16 fveq2 6375 . . . . . . . 8 (𝑘 = 𝑖 → (𝐵𝑘) = (𝐵𝑖))
1716oveq1d 6857 . . . . . . 7 (𝑘 = 𝑖 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑙)))
1815, 17eqeq12d 2780 . . . . . 6 (𝑘 = 𝑖 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))))
1913, 18imbi12d 335 . . . . 5 (𝑘 = 𝑖 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)))))
20 breq2 4813 . . . . . 6 (𝑙 = 𝑗 → (𝑖 < 𝑙𝑖 < 𝑗))
21 fveq2 6375 . . . . . . . 8 (𝑙 = 𝑗 → (𝐴𝑙) = (𝐴𝑗))
2221oveq2d 6858 . . . . . . 7 (𝑙 = 𝑗 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑗)))
23 fveq2 6375 . . . . . . . 8 (𝑙 = 𝑗 → (𝐵𝑙) = (𝐵𝑗))
2423oveq2d 6858 . . . . . . 7 (𝑙 = 𝑗 → ((𝐵𝑖) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑗)))
2522, 24eqeq12d 2780 . . . . . 6 (𝑙 = 𝑗 → (((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
2620, 25imbi12d 335 . . . . 5 (𝑙 = 𝑗 → ((𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))) ↔ (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
2719, 26cbvral2v 3327 . . . 4 (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
28 simpllr 793 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 ∈ dom 𝐴)
29 simplr 785 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑗 ∈ dom 𝐴)
30 simp-4r 803 . . . . . . . . . 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))))
3128, 29, 30jca31 510 . . . . . . . . 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)))))
32 simpr 477 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
3319, 26rspc2v 3474 . . . . . . . . . . 11 ((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
3433imp 395 . . . . . . . . . 10 (((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
3534imp 395 . . . . . . . . 9 ((((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 < 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
3631, 32, 35syl2anc 579 . . . . . . . 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)))
37 eqid 2765 . . . . . . . . . . 11 (Itv‘𝐺) = (Itv‘𝐺)
384ad3antrrr 721 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝐺 ∈ TarskiG)
396ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐴:𝐷𝑃)
40 simplr 785 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
4139fdmd 6232 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 = 𝐷)
4240, 41eleqtrd 2846 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖𝐷)
4339, 42ffvelrnd 6550 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑖) ∈ 𝑃)
4443adantr 472 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) ∈ 𝑃)
457ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐵:𝐷𝑃)
4645, 42ffvelrnd 6550 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑖) ∈ 𝑃)
4746adantr 472 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) ∈ 𝑃)
481, 2, 37, 38, 44, 47tgcgrtriv 25670 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐵𝑖) (𝐵𝑖)))
49 simpr 477 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
5049fveq2d 6379 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) = (𝐴𝑗))
5150oveq2d 6858 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐴𝑖) (𝐴𝑗)))
5249fveq2d 6379 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) = (𝐵𝑗))
5352oveq2d 6858 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐵𝑖) (𝐵𝑖)) = ((𝐵𝑖) (𝐵𝑗)))
5448, 51, 533eqtr3d 2807 . . . . . . . . 9 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
5554adantl3r 756 . . . . . . . 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)))
564ad4antr 724 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝐺 ∈ TarskiG)
57 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
5857, 41eleqtrd 2846 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗𝐷)
5939, 58ffvelrnd 6550 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑗) ∈ 𝑃)
6059adantr 472 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
6160adantl3r 756 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
6243adantr 472 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6362adantl3r 756 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6445, 58ffvelrnd 6550 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑗) ∈ 𝑃)
6564adantr 472 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6665adantl3r 756 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6746adantr 472 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
6867adantl3r 756 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
69 simplr 785 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 ∈ dom 𝐴)
70 simpllr 793 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑖 ∈ dom 𝐴)
71 simp-4r 803 . . . . . . . . . . 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))))
7269, 70, 71jca31 510 . . . . . . . . . 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)))))
73 simpr 477 . . . . . . . . . 10 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
74 breq1 4812 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑘 < 𝑙𝑗 < 𝑙))
75 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
7675oveq1d 6857 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑙)))
77 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐵𝑘) = (𝐵𝑗))
7877oveq1d 6857 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑙)))
7976, 78eqeq12d 2780 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))))
8074, 79imbi12d 335 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)))))
81 breq2 4813 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (𝑗 < 𝑙𝑗 < 𝑖))
82 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐴𝑙) = (𝐴𝑖))
8382oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑙 = 𝑖 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑖)))
84 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐵𝑙) = (𝐵𝑖))
8584oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑙 = 𝑖 → ((𝐵𝑗) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑖)))
8683, 85eqeq12d 2780 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
8781, 86imbi12d 335 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → ((𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))) ↔ (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))))
8880, 87rspc2v 3474 . . . . . . . . . . . 12 ((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))))
8988imp 395 . . . . . . . . . . 11 (((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
9089imp 395 . . . . . . . . . 10 ((((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑗 < 𝑖) → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))
9172, 73, 90syl2anc 579 . . . . . . . . 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)))
921, 2, 37, 56, 61, 63, 66, 68, 91tgcgrcomlr 25666 . . . . . . . 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)))
936fdmd 6232 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝐷)
9493, 5eqsstrd 3799 . . . . . . . . . . 11 (𝜑 → dom 𝐴 ⊆ ℝ)
9594ad3antrrr 721 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 ⊆ ℝ)
9640adantllr 710 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
9795, 96sseldd 3762 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ ℝ)
98 simpr 477 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
9995, 98sseldd 3762 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ ℝ)
10097, 99lttri4d 10432 . . . . . . . 8 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
10136, 55, 92, 100mpjao3dan 1556 . . . . . . 7 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
102101anasss 458 . . . . . 6 (((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
103102ralrimivva 3118 . . . . 5 ((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
104103ex 401 . . . 4 (𝜑 → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10527, 104syl5bir 234 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10612, 105impbid 203 . 2 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1078, 106bitrd 270 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732   class class class wbr 4809  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  cr 10188   < clt 10328  Basecbs 16130  distcds 16223  TarskiGcstrkg 25620  Itvcitv 25626  cgrGccgrg 25696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-pre-lttri 10263  ax-pre-lttrn 10264
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-er 7947  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-ltxr 10333  df-trkgc 25638  df-trkgcb 25640  df-trkg 25643  df-cgrg 25697
This theorem is referenced by:  tgcgr4  25717
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