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Theorem iscgrglt 25322
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 25321 . 2 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
9 simp2 1060 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) ∧ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ∧ 𝑖 < 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
1093expia 1264 . . . . 5 (((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) ∧ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
1110ex 450 . . . 4 ((𝜑 ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → (((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1211ralimdvva 2959 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
13 breq1 4621 . . . . . 6 (𝑘 = 𝑖 → (𝑘 < 𝑙𝑖 < 𝑙))
14 fveq2 6153 . . . . . . . 8 (𝑘 = 𝑖 → (𝐴𝑘) = (𝐴𝑖))
1514oveq1d 6625 . . . . . . 7 (𝑘 = 𝑖 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑙)))
16 fveq2 6153 . . . . . . . 8 (𝑘 = 𝑖 → (𝐵𝑘) = (𝐵𝑖))
1716oveq1d 6625 . . . . . . 7 (𝑘 = 𝑖 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑙)))
1815, 17eqeq12d 2636 . . . . . 6 (𝑘 = 𝑖 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))))
1913, 18imbi12d 334 . . . . 5 (𝑘 = 𝑖 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)))))
20 breq2 4622 . . . . . 6 (𝑙 = 𝑗 → (𝑖 < 𝑙𝑖 < 𝑗))
21 fveq2 6153 . . . . . . . 8 (𝑙 = 𝑗 → (𝐴𝑙) = (𝐴𝑗))
2221oveq2d 6626 . . . . . . 7 (𝑙 = 𝑗 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐴𝑖) (𝐴𝑗)))
23 fveq2 6153 . . . . . . . 8 (𝑙 = 𝑗 → (𝐵𝑙) = (𝐵𝑗))
2423oveq2d 6626 . . . . . . 7 (𝑙 = 𝑗 → ((𝐵𝑖) (𝐵𝑙)) = ((𝐵𝑖) (𝐵𝑗)))
2522, 24eqeq12d 2636 . . . . . 6 (𝑙 = 𝑗 → (((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
2620, 25imbi12d 334 . . . . 5 (𝑙 = 𝑗 → ((𝑖 < 𝑙 → ((𝐴𝑖) (𝐴𝑙)) = ((𝐵𝑖) (𝐵𝑙))) ↔ (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
2719, 26cbvral2v 3170 . . . 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 798 . . . . . . . . . 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 791 . . . . . . . . . 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 806 . . . . . . . . . 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 556 . . . . . . . . 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 3310 . . . . . . . . . . 11 ((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
3433imp 445 . . . . . . . . . 10 (((𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
3534imp 445 . . . . . . . . 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 692 . . . . . . . 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 2621 . . . . . . . . . . 11 (Itv‘𝐺) = (Itv‘𝐺)
384ad3antrrr 765 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝐺 ∈ TarskiG)
396ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐴:𝐷𝑃)
40 simplr 791 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
41 fdm 6013 . . . . . . . . . . . . . . 15 (𝐴:𝐷𝑃 → dom 𝐴 = 𝐷)
4239, 41syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 = 𝐷)
4340, 42eleqtrd 2700 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖𝐷)
4439, 43ffvelrnd 6321 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑖) ∈ 𝑃)
4544adantr 481 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) ∈ 𝑃)
467ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝐵:𝐷𝑃)
4746, 43ffvelrnd 6321 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑖) ∈ 𝑃)
4847adantr 481 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) ∈ 𝑃)
491, 2, 37, 38, 45, 48tgcgrtriv 25292 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐵𝑖) (𝐵𝑖)))
50 simpr 477 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
5150fveq2d 6157 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐴𝑖) = (𝐴𝑗))
5251oveq2d 6626 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑖)) = ((𝐴𝑖) (𝐴𝑗)))
5350fveq2d 6157 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → (𝐵𝑖) = (𝐵𝑗))
5453oveq2d 6626 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐵𝑖) (𝐵𝑖)) = ((𝐵𝑖) (𝐵𝑗)))
5549, 52, 543eqtr3d 2663 . . . . . . . . 9 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑖 = 𝑗) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
5655adantl3r 785 . . . . . . . 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)))
574ad4antr 767 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝐺 ∈ TarskiG)
58 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
5958, 42eleqtrd 2700 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗𝐷)
6039, 59ffvelrnd 6321 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐴𝑗) ∈ 𝑃)
6160adantr 481 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
6261adantl3r 785 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑗) ∈ 𝑃)
6344adantr 481 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6463adantl3r 785 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐴𝑖) ∈ 𝑃)
6546, 59ffvelrnd 6321 . . . . . . . . . . 11 (((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝐵𝑗) ∈ 𝑃)
6665adantr 481 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6766adantl3r 785 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑗) ∈ 𝑃)
6847adantr 481 . . . . . . . . . 10 ((((𝜑𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
6968adantl3r 785 . . . . . . . . 9 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → (𝐵𝑖) ∈ 𝑃)
70 simplr 791 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑗 ∈ dom 𝐴)
71 simpllr 798 . . . . . . . . . . 11 (((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) ∧ 𝑗 < 𝑖) → 𝑖 ∈ dom 𝐴)
72 simp-4r 806 . . . . . . . . . . 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))))
7370, 71, 72jca31 556 . . . . . . . . . 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)))))
74 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)
75 breq1 4621 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑘 < 𝑙𝑗 < 𝑙))
76 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐴𝑘) = (𝐴𝑗))
7776oveq1d 6625 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑙)))
78 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐵𝑘) = (𝐵𝑗))
7978oveq1d 6625 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → ((𝐵𝑘) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑙)))
8077, 79eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))))
8175, 80imbi12d 334 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) ↔ (𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)))))
82 breq2 4622 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (𝑗 < 𝑙𝑗 < 𝑖))
83 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐴𝑙) = (𝐴𝑖))
8483oveq2d 6626 . . . . . . . . . . . . . . 15 (𝑙 = 𝑖 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐴𝑗) (𝐴𝑖)))
85 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐵𝑙) = (𝐵𝑖))
8685oveq2d 6626 . . . . . . . . . . . . . . 15 (𝑙 = 𝑖 → ((𝐵𝑗) (𝐵𝑙)) = ((𝐵𝑗) (𝐵𝑖)))
8784, 86eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙)) ↔ ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
8882, 87imbi12d 334 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → ((𝑗 < 𝑙 → ((𝐴𝑗) (𝐴𝑙)) = ((𝐵𝑗) (𝐵𝑙))) ↔ (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))))
8981, 88rspc2v 3310 . . . . . . . . . . . 12 ((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))))
9089imp 445 . . . . . . . . . . 11 (((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → (𝑗 < 𝑖 → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖))))
9190imp 445 . . . . . . . . . 10 ((((𝑗 ∈ dom 𝐴𝑖 ∈ dom 𝐴) ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑗 < 𝑖) → ((𝐴𝑗) (𝐴𝑖)) = ((𝐵𝑗) (𝐵𝑖)))
9273, 74, 91syl2anc 692 . . . . . . . . 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)))
931, 2, 37, 57, 62, 64, 67, 69, 92tgcgrcomlr 25288 . . . . . . . 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)))
946, 41syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝐴 = 𝐷)
9594, 5eqsstrd 3623 . . . . . . . . . . 11 (𝜑 → dom 𝐴 ⊆ ℝ)
9695ad3antrrr 765 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → dom 𝐴 ⊆ ℝ)
9740adantllr 754 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ dom 𝐴)
9896, 97sseldd 3588 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑖 ∈ ℝ)
99 simpr 477 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ dom 𝐴)
10096, 99sseldd 3588 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → 𝑗 ∈ ℝ)
10198, 100lttri4d 10129 . . . . . . . 8 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
10236, 56, 93, 101mpjao3dan 1392 . . . . . . 7 ((((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ 𝑖 ∈ dom 𝐴) ∧ 𝑗 ∈ dom 𝐴) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
103102anasss 678 . . . . . 6 (((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) ∧ (𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴)) → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
104103ralrimivva 2966 . . . . 5 ((𝜑 ∧ ∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙)))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
105104ex 450 . . . 4 (𝜑 → (∀𝑘 ∈ dom 𝐴𝑙 ∈ dom 𝐴(𝑘 < 𝑙 → ((𝐴𝑘) (𝐴𝑙)) = ((𝐵𝑘) (𝐵𝑙))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10627, 105syl5bir 233 . . 3 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) → ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
10712, 106impbid 202 . 2 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
1088, 107bitrd 268 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559   class class class wbr 4618  dom cdm 5079  wf 5848  cfv 5852  (class class class)co 6610  cr 9886   < clt 10025  Basecbs 15788  distcds 15878  TarskiGcstrkg 25242  Itvcitv 25248  cgrGccgrg 25318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-pre-lttri 9961  ax-pre-lttrn 9962
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-er 7694  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-ltxr 10030  df-trkgc 25260  df-trkgcb 25262  df-trkg 25265  df-cgrg 25319
This theorem is referenced by:  tgcgr4  25339
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