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Theorem abvpropd 20301
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 20006 . . . 4 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61, 2eqtr3d 2778 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
76feq2d 6654 . . . . 5 (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞)))
81, 2, 3grpidpropd 18517 . . . . . . . . . . 11 (𝜑 → (0g𝐾) = (0g𝐿))
98adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (0g𝐾) = (0g𝐿))
109eqeq2d 2747 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥 = (0g𝐾) ↔ 𝑥 = (0g𝐿)))
1110bibi2d 342 . . . . . . . 8 ((𝜑𝑥𝐵) → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿))))
124fveqeq2d 6850 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
133fveq2d 6846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐾)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
1413breq1d 5115 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
1512, 14anbi12d 631 . . . . . . . . . 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 468 . . . . . . . . 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 3172 . . . . . . . 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 631 . . . . . . 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 3172 . . . . . 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 3313 . . . . . . . 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 629 . . . . . . 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 3319 . . . . . 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 3313 . . . . . . . 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 629 . . . . . . 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 3319 . . . . . 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 308 . . . . 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 631 . . . 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 631 . . 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 2736 . . . . 5 (AbsVal‘𝐾) = (AbsVal‘𝐾)
3029abvrcl 20280 . . . 4 (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring)
31 eqid 2736 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2736 . . . . 5 (+g𝐾) = (+g𝐾)
33 eqid 2736 . . . . 5 (.r𝐾) = (.r𝐾)
34 eqid 2736 . . . . 5 (0g𝐾) = (0g𝐾)
3529, 31, 32, 33, 34isabv 20278 . . . 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 820 . . 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 2736 . . . . 5 (AbsVal‘𝐿) = (AbsVal‘𝐿)
3837abvrcl 20280 . . . 4 (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring)
39 eqid 2736 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
40 eqid 2736 . . . . 5 (+g𝐿) = (+g𝐿)
41 eqid 2736 . . . . 5 (.r𝐿) = (.r𝐿)
42 eqid 2736 . . . . 5 (0g𝐿) = (0g𝐿)
4337, 39, 40, 41, 42isabv 20278 . . . 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 820 . . 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 313 . 2 (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿)))
4645eqrdv 2734 1 (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051   + caddc 11054   · cmul 11056  +∞cpnf 11186  cle 11190  [,)cico 13266  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  Ringcrg 19964  AbsValcabv 20275
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-mgp 19897  df-ring 19966  df-abv 20276
This theorem is referenced by:  tngnrg  24038  abvpropd2  31819
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