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Theorem unxpwdom3 39688
Description: Weaker version of unxpwdom 9047 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)
31, 2xpexd 7468 . 2 (𝜑 → (𝐷 × 𝐵) ∈ V)
4 simprr 771 . . . . 5 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (𝑎 + 𝑑) ∈ 𝐵)
5 simplr 767 . . . . . . 7 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎𝐶)
6 unxpwdom3.rc . . . . . . . . . 10 (((𝜑𝑑𝐷) ∧ (𝑎𝐶𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
76an4s 658 . . . . . . . . 9 (((𝜑𝑎𝐶) ∧ (𝑑𝐷𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
87anassrs 470 . . . . . . . 8 ((((𝜑𝑎𝐶) ∧ 𝑑𝐷) ∧ 𝑐𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
98adantlrr 719 . . . . . . 7 ((((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) ∧ 𝑐𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
105, 9riota5 7137 . . . . . 6 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)) = 𝑎)
1110eqcomd 2827 . . . . 5 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
12 eqeq2 2833 . . . . . . 7 (𝑦 = (𝑎 + 𝑑) → ((𝑐 + 𝑑) = 𝑦 ↔ (𝑐 + 𝑑) = (𝑎 + 𝑑)))
1312riotabidv 7110 . . . . . 6 (𝑦 = (𝑎 + 𝑑) → (𝑐𝐶 (𝑐 + 𝑑) = 𝑦) = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
1413rspceeqv 3638 . . . . 5 (((𝑎 + 𝑑) ∈ 𝐵𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑))) → ∃𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
154, 11, 14syl2anc 586 . . . 4 (((𝜑𝑎𝐶) ∧ (𝑑𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → ∃𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
16 unxpwdom3.ni . . . . . . 7 (𝜑 → ¬ 𝐷𝐴)
1716adantr 483 . . . . . 6 ((𝜑𝑎𝐶) → ¬ 𝐷𝐴)
18 unxpwdom3.av . . . . . . . 8 (𝜑𝐴𝑉)
1918ad2antrr 724 . . . . . . 7 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐴𝑉)
20 oveq2 7158 . . . . . . . . . . . . . 14 (𝑑 = 𝑏 → (𝑎 + 𝑑) = (𝑎 + 𝑏))
2120eleq1d 2897 . . . . . . . . . . . . 13 (𝑑 = 𝑏 → ((𝑎 + 𝑑) ∈ 𝐵 ↔ (𝑎 + 𝑏) ∈ 𝐵))
2221notbid 320 . . . . . . . . . . . 12 (𝑑 = 𝑏 → (¬ (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ (𝑎 + 𝑏) ∈ 𝐵))
2322rspcv 3618 . . . . . . . . . . 11 (𝑏𝐷 → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
2423adantl 484 . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
25 unxpwdom3.ov . . . . . . . . . . . . . 14 ((𝜑𝑎𝐶𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))
26253expa 1114 . . . . . . . . . . . . 13 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))
27 elun 4125 . . . . . . . . . . . . 13 ((𝑎 + 𝑏) ∈ (𝐴𝐵) ↔ ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
2826, 27sylib 220 . . . . . . . . . . . 12 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
2928orcomd 867 . . . . . . . . . . 11 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → ((𝑎 + 𝑏) ∈ 𝐵 ∨ (𝑎 + 𝑏) ∈ 𝐴))
3029ord 860 . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (¬ (𝑎 + 𝑏) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
3124, 30syld 47 . . . . . . . . 9 (((𝜑𝑎𝐶) ∧ 𝑏𝐷) → (∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
3231impancom 454 . . . . . . . 8 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝑏𝐷 → (𝑎 + 𝑏) ∈ 𝐴))
33 unxpwdom3.lc . . . . . . . . . 10 (((𝜑𝑎𝐶) ∧ (𝑏𝐷𝑐𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))
3433ex 415 . . . . . . . . 9 ((𝜑𝑎𝐶) → ((𝑏𝐷𝑐𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
3534adantr 483 . . . . . . . 8 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → ((𝑏𝐷𝑐𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
3632, 35dom2d 8544 . . . . . . 7 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝐴𝑉𝐷𝐴))
3719, 36mpd 15 . . . . . 6 (((𝜑𝑎𝐶) ∧ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐷𝐴)
3817, 37mtand 814 . . . . 5 ((𝜑𝑎𝐶) → ¬ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
39 dfrex2 3239 . . . . 5 (∃𝑑𝐷 (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ ∀𝑑𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
4038, 39sylibr 236 . . . 4 ((𝜑𝑎𝐶) → ∃𝑑𝐷 (𝑎 + 𝑑) ∈ 𝐵)
4115, 40reximddv 3275 . . 3 ((𝜑𝑎𝐶) → ∃𝑑𝐷𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
42 vex 3498 . . . . . . . . 9 𝑑 ∈ V
43 vex 3498 . . . . . . . . 9 𝑦 ∈ V
4442, 43op1std 7693 . . . . . . . 8 (𝑥 = ⟨𝑑, 𝑦⟩ → (1st𝑥) = 𝑑)
4544oveq2d 7166 . . . . . . 7 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑐 + (1st𝑥)) = (𝑐 + 𝑑))
4642, 43op2ndd 7694 . . . . . . 7 (𝑥 = ⟨𝑑, 𝑦⟩ → (2nd𝑥) = 𝑦)
4745, 46eqeq12d 2837 . . . . . 6 (𝑥 = ⟨𝑑, 𝑦⟩ → ((𝑐 + (1st𝑥)) = (2nd𝑥) ↔ (𝑐 + 𝑑) = 𝑦))
4847riotabidv 7110 . . . . 5 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
4948eqeq2d 2832 . . . 4 (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) ↔ 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦)))
5049rexxp 5708 . . 3 (∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)) ↔ ∃𝑑𝐷𝑦𝐵 𝑎 = (𝑐𝐶 (𝑐 + 𝑑) = 𝑦))
5141, 50sylibr 236 . 2 ((𝜑𝑎𝐶) → ∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (𝑐𝐶 (𝑐 + (1st𝑥)) = (2nd𝑥)))
523, 51wdomd 9039 1 (𝜑𝐶* (𝐷 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3495  cun 3934  cop 4567   class class class wbr 5059   × cxp 5548  cfv 6350  crio 7107  (class class class)co 7150  1st c1st 7681  2nd c2nd 7682  cdom 8501  * cwdom 9015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-1st 7683  df-2nd 7684  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-wdom 9017
This theorem is referenced by:  isnumbasgrplem2  39697
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