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Theorem abvpropd 18607
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 18347 . . . 4 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61, 2eqtr3d 2641 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
76feq2d 5926 . . . . 5 (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞)))
81, 2, 3grpidpropd 17026 . . . . . . . . . . 11 (𝜑 → (0g𝐾) = (0g𝐿))
98adantr 479 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (0g𝐾) = (0g𝐿))
109eqeq2d 2615 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥 = (0g𝐾) ↔ 𝑥 = (0g𝐿)))
1110bibi2d 330 . . . . . . . 8 ((𝜑𝑥𝐵) → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿))))
124fveq2d 6088 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(.r𝐾)𝑦)) = (𝑓‘(𝑥(.r𝐿)𝑦)))
1312eqeq1d 2607 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
143fveq2d 6088 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐾)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
1514breq1d 4583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
1613, 15anbi12d 742 . . . . . . . . . 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 677 . . . . . . . . 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 2963 . . . . . . . 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 742 . . . . . . 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 2963 . . . . . 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 3116 . . . . . . . 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 735 . . . . . . 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 3124 . . . . . 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 3116 . . . . . . . 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 735 . . . . . . 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 3124 . . . . . 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 296 . . . . 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 742 . . . 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 742 . . 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 2605 . . . . 5 (AbsVal‘𝐾) = (AbsVal‘𝐾)
3130abvrcl 18586 . . . 4 (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring)
32 eqid 2605 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
33 eqid 2605 . . . . 5 (+g𝐾) = (+g𝐾)
34 eqid 2605 . . . . 5 (.r𝐾) = (.r𝐾)
35 eqid 2605 . . . . 5 (0g𝐾) = (0g𝐾)
3630, 32, 33, 34, 35isabv 18584 . . . 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 671 . . 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 2605 . . . . 5 (AbsVal‘𝐿) = (AbsVal‘𝐿)
3938abvrcl 18586 . . . 4 (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring)
40 eqid 2605 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2605 . . . . 5 (+g𝐿) = (+g𝐿)
42 eqid 2605 . . . . 5 (.r𝐿) = (.r𝐿)
43 eqid 2605 . . . . 5 (0g𝐿) = (0g𝐿)
4438, 40, 41, 42, 43isabv 18584 . . . 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 671 . . 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 301 . 2 (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿)))
4746eqrdv 2603 1 (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891   class class class wbr 4573  wf 5782  cfv 5786  (class class class)co 6523  0cc0 9788   + caddc 9791   · cmul 9793  +∞cpnf 9923  cle 9927  [,)cico 12000  Basecbs 15637  +gcplusg 15710  .rcmulr 15711  0gc0g 15865  Ringcrg 18312  AbsValcabv 18581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-er 7602  df-map 7719  df-en 7815  df-dom 7816  df-sdom 7817  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-plusg 15723  df-0g 15867  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-grp 17190  df-mgp 18255  df-ring 18314  df-abv 18582
This theorem is referenced by:  tngnrg  22217  abvpropd2  28785
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