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Theorem abvpropd 19338
Description: If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
abvpropd.1 (𝜑𝐵 = (Base‘𝐾))
abvpropd.2 (𝜑𝐵 = (Base‘𝐿))
abvpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
abvpropd.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
abvpropd (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem abvpropd
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 abvpropd.1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
2 abvpropd.2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
3 abvpropd.3 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
4 abvpropd.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
51, 2, 3, 4ringpropd 19058 . . . 4 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61, 2eqtr3d 2816 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
76feq2d 6332 . . . . 5 (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞)))
81, 2, 3grpidpropd 17732 . . . . . . . . . . 11 (𝜑 → (0g𝐾) = (0g𝐿))
98adantr 473 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (0g𝐾) = (0g𝐿))
109eqeq2d 2788 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥 = (0g𝐾) ↔ 𝑥 = (0g𝐿)))
1110bibi2d 335 . . . . . . . 8 ((𝜑𝑥𝐵) → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿))))
124fveqeq2d 6509 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
133fveq2d 6505 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐾)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
1413breq1d 4940 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
1512, 14anbi12d 621 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1615anassrs 460 . . . . . . . . 9 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1716ralbidva 3146 . . . . . . . 8 ((𝜑𝑥𝐵) → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1811, 17anbi12d 621 . . . . . . 7 ((𝜑𝑥𝐵) → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
1918ralbidva 3146 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
201raleqdv 3355 . . . . . . . 8 (𝜑 → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
2120anbi2d 619 . . . . . . 7 (𝜑 → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
221, 21raleqbidv 3341 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
232raleqdv 3355 . . . . . . . 8 (𝜑 → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
2423anbi2d 619 . . . . . . 7 (𝜑 → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
252, 24raleqbidv 3341 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
2619, 22, 253bitr3d 301 . . . . 5 (𝜑 → (∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
277, 26anbi12d 621 . . . 4 (𝜑 → ((𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
285, 27anbi12d 621 . . 3 (𝜑 → ((𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))))
29 eqid 2778 . . . . 5 (AbsVal‘𝐾) = (AbsVal‘𝐾)
3029abvrcl 19317 . . . 4 (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring)
31 eqid 2778 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2778 . . . . 5 (+g𝐾) = (+g𝐾)
33 eqid 2778 . . . . 5 (.r𝐾) = (.r𝐾)
34 eqid 2778 . . . . 5 (0g𝐾) = (0g𝐾)
3529, 31, 32, 33, 34isabv 19315 . . . 4 (𝐾 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
3630, 35biadanii 810 . . 3 (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
37 eqid 2778 . . . . 5 (AbsVal‘𝐿) = (AbsVal‘𝐿)
3837abvrcl 19317 . . . 4 (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring)
39 eqid 2778 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
40 eqid 2778 . . . . 5 (+g𝐿) = (+g𝐿)
41 eqid 2778 . . . . 5 (.r𝐿) = (.r𝐿)
42 eqid 2778 . . . . 5 (0g𝐿) = (0g𝐿)
4337, 39, 40, 41, 42isabv 19315 . . . 4 (𝐿 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
4438, 43biadanii 810 . . 3 (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
4528, 36, 443bitr4g 306 . 2 (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿)))
4645eqrdv 2776 1 (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3088   class class class wbr 4930  wf 6186  cfv 6190  (class class class)co 6978  0cc0 10337   + caddc 10340   · cmul 10342  +∞cpnf 10473  cle 10477  [,)cico 12559  Basecbs 16342  +gcplusg 16424  .rcmulr 16425  0gc0g 16572  Ringcrg 19023  AbsValcabv 19312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-er 8091  df-map 8210  df-en 8309  df-dom 8310  df-sdom 8311  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-2 11506  df-ndx 16345  df-slot 16346  df-base 16348  df-sets 16349  df-plusg 16437  df-0g 16574  df-mgm 17713  df-sgrp 17755  df-mnd 17766  df-grp 17897  df-mgp 18966  df-ring 19025  df-abv 19313
This theorem is referenced by:  tngnrg  22989  abvpropd2  30371
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