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Theorem abvpropd 18665
 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 18405 . . . 4 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61, 2eqtr3d 2646 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
76feq2d 5944 . . . . 5 (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞)))
81, 2, 3grpidpropd 17084 . . . . . . . . . . 11 (𝜑 → (0g𝐾) = (0g𝐿))
98adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (0g𝐾) = (0g𝐿))
109eqeq2d 2620 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥 = (0g𝐾) ↔ 𝑥 = (0g𝐿)))
1110bibi2d 331 . . . . . . . 8 ((𝜑𝑥𝐵) → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿))))
124fveq2d 6107 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(.r𝐾)𝑦)) = (𝑓‘(𝑥(.r𝐿)𝑦)))
1312eqeq1d 2612 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
143fveq2d 6107 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐾)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
1514breq1d 4593 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
1613, 15anbi12d 743 . . . . . . . . . 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)))))
1716anassrs 678 . . . . . . . . 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)))))
1817ralbidva 2968 . . . . . . . 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)))))
1911, 18anbi12d 743 . . . . . . 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))))))
2019ralbidva 2968 . . . . . 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))))))
211raleqdv 3121 . . . . . . . 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)))))
2221anbi2d 736 . . . . . . 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))))))
231, 22raleqbidv 3129 . . . . . 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))))))
242raleqdv 3121 . . . . . . . 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)))))
2524anbi2d 736 . . . . . . 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))))))
262, 25raleqbidv 3129 . . . . . 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))))))
2720, 23, 263bitr3d 297 . . . . 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))))))
287, 27anbi12d 743 . . . 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)))))))
295, 28anbi12d 743 . . 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))))))))
30 eqid 2610 . . . . 5 (AbsVal‘𝐾) = (AbsVal‘𝐾)
3130abvrcl 18644 . . . 4 (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring)
32 eqid 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
33 eqid 2610 . . . . 5 (+g𝐾) = (+g𝐾)
34 eqid 2610 . . . . 5 (.r𝐾) = (.r𝐾)
35 eqid 2610 . . . . 5 (0g𝐾) = (0g𝐾)
3630, 32, 33, 34, 35isabv 18642 . . . 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)))))))
3731, 36biadan2 672 . . 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)))))))
38 eqid 2610 . . . . 5 (AbsVal‘𝐿) = (AbsVal‘𝐿)
3938abvrcl 18644 . . . 4 (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring)
40 eqid 2610 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2610 . . . . 5 (+g𝐿) = (+g𝐿)
42 eqid 2610 . . . . 5 (.r𝐿) = (.r𝐿)
43 eqid 2610 . . . . 5 (0g𝐿) = (0g𝐿)
4438, 40, 41, 42, 43isabv 18642 . . . 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)))))))
4539, 44biadan2 672 . . 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)))))))
4629, 37, 453bitr4g 302 . 2 (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿)))
4746eqrdv 2608 1 (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818   · cmul 9820  +∞cpnf 9950   ≤ cle 9954  [,)cico 12048  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Ringcrg 18370  AbsValcabv 18639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-mgp 18313  df-ring 18372  df-abv 18640 This theorem is referenced by:  tngnrg  22288  abvpropd2  28983
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