Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unxpwdom3 Structured version   Visualization version   GIF version

Theorem unxpwdom3 36573
Description: Weaker version of unxpwdom 8252 where a function is required only to be cancellative, not an injection. 𝐷 and 𝐵 are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into 𝐴, each row must hit an element of 𝐵; by column injectivity, each row can be identified in at least one way by the 𝐵 element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
Hypotheses
Ref Expression
unxpwdom3.av (𝜑𝐴𝑉)
unxpwdom3.bv (𝜑𝐵𝑊)
unxpwdom3.dv (𝜑𝐷𝑋)
unxpwdom3.ov ((𝜑𝑎𝐶𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))
unxpwdom3.lc (((𝜑𝑎𝐶) ∧ (𝑏𝐷𝑐𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))
unxpwdom3.rc (((𝜑𝑑𝐷) ∧ (𝑎𝐶𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
unxpwdom3.ni (𝜑 → ¬ 𝐷𝐴)
Assertion
Ref Expression
unxpwdom3 (𝜑𝐶* (𝐷 × 𝐵))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝐵   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑   + ,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏,𝑐,𝑑   𝐴,𝑏,𝑐
Allowed substitution hints:   𝐴(𝑎,𝑑)   𝑉(𝑎,𝑏,𝑐,𝑑)   𝑊(𝑎,𝑏,𝑐,𝑑)   𝑋(𝑎,𝑏,𝑐,𝑑)

Proof of Theorem unxpwdom3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unxpwdom3.dv . . 3 (𝜑𝐷𝑋)
2 unxpwdom3.bv . . 3 (𝜑𝐵𝑊)
3 xpexg 6733 . . 3 ((𝐷𝑋𝐵𝑊) → (𝐷 × 𝐵) ∈ V)
41, 2, 3syl2anc 690 . 2 (𝜑 → (𝐷 × 𝐵) ∈ V)
5 simprr 791 . . . . 5 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (𝑎 + 𝑑) ∈ 𝐵)
6 simplr 787 . . . . . . 7 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎𝐶)
7 unxpwdom3.rc . . . . . . . . . 10 (((𝜑𝑑𝐷) ∧ (𝑎𝐶𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
87an4s 864 . . . . . . . . 9 (((𝜑𝑎𝐶) ∧ (𝑑𝐷𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
98anassrs 677 . . . . . . . 8 ((((𝜑𝑎𝐶) ∧ 𝑑𝐷) ∧ 𝑐𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
109adantlrr 752 . . . . . . 7 ((((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) ∧ 𝑐𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
116, 10riota5 6412 . . . . . 6 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)) = 𝑎)
1211eqcomd 2520 . . . . 5 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
13 eqeq2 2525 . . . . . . . 8 (𝑦 = (𝑎 + 𝑑) → ((𝑐 + 𝑑) = 𝑦 ↔ (𝑐 + 𝑑) = (𝑎 + 𝑑)))
1413riotabidv 6389 . . . . . . 7 (𝑦 = (𝑎 + 𝑑) → (𝑐𝐶 (𝑐 + 𝑑) = 𝑦) = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
1514eqeq2d 2524 . . . . . 6 (𝑦 = (𝑎 + 𝑑) → (𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦) ↔ 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑))))
1615rspcev 3186 . . . . 5 (((𝑎 + 𝑑) ∈ 𝐵𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑))) → ∃𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
175, 12, 16syl2anc 690 . . . 4 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → ∃𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
18 unxpwdom3.ni . . . . . . 7 (𝜑 → ¬ 𝐷𝐴)
1918adantr 479 . . . . . 6 ((𝜑𝑎𝐶) → ¬ 𝐷𝐴)
20 unxpwdom3.av . . . . . . . 8 (𝜑𝐴𝑉)
2120ad2antrr 757 . . . . . . 7 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐴𝑉)
22 oveq2 6433 . . . . . . . . . . . . . 14 (𝑑 = 𝑏 → (𝑎 + 𝑑) = (𝑎 + 𝑏))
2322eleq1d 2576 . . . . . . . . . . . . 13 (𝑑 = 𝑏 → ((𝑎 + 𝑑) ∈ 𝐵 ↔ (𝑎 + 𝑏) ∈ 𝐵))
2423notbid 306 . . . . . . . . . . . 12 (𝑑 = 𝑏 → (¬ (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ (𝑎 + 𝑏) ∈ 𝐵))
2524rspcv 3182 . . . . . . . . . . 11 (𝑏𝐷 → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
2625adantl 480 . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
27 unxpwdom3.ov . . . . . . . . . . . . . 14 ((𝜑𝑎𝐶𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))
28273expa 1256 . . . . . . . . . . . . 13 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))
29 elun 3619 . . . . . . . . . . . . 13 ((𝑎 + 𝑏) ∈ (𝐴𝐵) ↔ ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
3028, 29sylib 206 . . . . . . . . . . . 12 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
3130orcomd 401 . . . . . . . . . . 11 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → ((𝑎 + 𝑏) ∈ 𝐵 ∨ (𝑎 + 𝑏) ∈ 𝐴))
3231ord 390 . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (¬ (𝑎 + 𝑏) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
3326, 32syld 45 . . . . . . . . 9 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
3433impancom 454 . . . . . . . 8 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝑏𝐷 → (𝑎 + 𝑏) ∈ 𝐴))
35 unxpwdom3.lc . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ (𝑏𝐷𝑐𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))
3635ex 448 . . . . . . . . 9 ((𝜑𝑎𝐶) → ((𝑏𝐷𝑐𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
3736adantr 479 . . . . . . . 8 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → ((𝑏𝐷𝑐𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
3834, 37dom2d 7757 . . . . . . 7 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝐴𝑉𝐷𝐴))
3921, 38mpd 15 . . . . . 6 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐷𝐴)
4019, 39mtand 688 . . . . 5 ((𝜑𝑎𝐶) → ¬ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
41 dfrex2 2883 . . . . 5 (∃𝑑𝐷 (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
4240, 41sylibr 222 . . . 4 ((𝜑𝑎𝐶) → ∃𝑑𝐷 (𝑎 + 𝑑) ∈ 𝐵)
4317, 42reximddv 2905 . . 3 ((𝜑𝑎𝐶) → ∃𝑑𝐷𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
44 vex 3080 . . . . . . . . 9 𝑑 ∈ V
45 vex 3080 . . . . . . . . 9 𝑦 ∈ V
4644, 45op1std 6943 . . . . . . . 8 (𝑥 = ⟨𝑑, 𝑦⟩ → (1st𝑥) = 𝑑)
4746oveq2d 6441 . . . . . . 7 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑐 + (1st𝑥)) = (𝑐 + 𝑑))
4844, 45op2ndd 6944 . . . . . . 7 (𝑥 = ⟨𝑑, 𝑦⟩ → (2nd𝑥) = 𝑦)
4947, 48eqeq12d 2529 . . . . . 6 (𝑥 = ⟨𝑑, 𝑦⟩ → ((𝑐 + (1st𝑥)) = (2nd𝑥) ↔ (𝑐 + 𝑑) = 𝑦))
5049riotabidv 6389 . . . . 5 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
5150eqeq2d 2524 . . . 4 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) ↔ 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦)))
5251rexxp 5078 . . 3 (∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) ↔ ∃𝑑𝐷𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
5343, 52sylibr 222 . 2 ((𝜑𝑎𝐶) → ∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)))
544, 53wdomd 8244 1 (𝜑𝐶* (𝐷 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  wrex 2801  Vcvv 3077  cun 3442  cop 4034   class class class wbr 4481   × cxp 4930  cfv 5689  crio 6386  (class class class)co 6425  1st c1st 6931  2nd c2nd 6932  cdom 7714  * cwdom 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-1st 6933  df-2nd 6934  df-er 7504  df-en 7717  df-dom 7718  df-sdom 7719  df-wdom 8222
This theorem is referenced by:  isnumbasgrplem2  36583
  Copyright terms: Public domain W3C validator