Theorem unxpwdom3 36683

Description: Weaker version of unxpwdom 8377 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	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unxpwdom3.bv	⊢ (𝜑 → 𝐵 ∈ 𝑊)
unxpwdom3.dv	⊢ (𝜑 → 𝐷 ∈ 𝑋)
unxpwdom3.ov	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐷) → (𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵))
unxpwdom3.lc	⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))
unxpwdom3.rc	⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ (𝑎 ∈ 𝐶 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
unxpwdom3.ni	⊢ (𝜑 → ¬ 𝐷 ≼ 𝐴)

Assertion

Ref	Expression
unxpwdom3	⊢ (𝜑 → 𝐶 ≼* (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.

Step	Hyp	Ref	Expression
1		unxpwdom3.dv	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
2		unxpwdom3.bv	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		xpexg 6858	. . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝐷 × 𝐵) ∈ V)
4	1, 2, 3	syl2anc 691	. 2 ⊢ (𝜑 → (𝐷 × 𝐵) ∈ V)
5		simprr 792	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (𝑎 + 𝑑) ∈ 𝐵)
6		simplr 788	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎 ∈ 𝐶)
7		unxpwdom3.rc	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ (𝑎 ∈ 𝐶 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
8	7	an4s 865	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
9	8	anassrs 678	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑑 ∈ 𝐷) ∧ 𝑐 ∈ 𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
10	9	adantlrr 753	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) ∧ 𝑐 ∈ 𝐶) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))
11	6, 10	riota5 6536	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)) = 𝑎)
12	11	eqcomd 2616	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
13		eqeq2 2621	. . . . . . . 8 ⊢ (𝑦 = (𝑎 + 𝑑) → ((𝑐 + 𝑑) = 𝑦 ↔ (𝑐 + 𝑑) = (𝑎 + 𝑑)))
14	13	riotabidv 6513	. . . . . . 7 ⊢ (𝑦 = (𝑎 + 𝑑) → (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦) = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑)))
15	14	eqeq2d 2620	. . . . . 6 ⊢ (𝑦 = (𝑎 + 𝑑) → (𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦) ↔ 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑))))
16	15	rspcev 3282	. . . . 5 ⊢ (((𝑎 + 𝑑) ∈ 𝐵 ∧ 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = (𝑎 + 𝑑))) → ∃𝑦 ∈ 𝐵 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦))
17	5, 12, 16	syl2anc 691	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑑 ∈ 𝐷 ∧ (𝑎 + 𝑑) ∈ 𝐵)) → ∃𝑦 ∈ 𝐵 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦))
18		unxpwdom3.ni	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐷 ≼ 𝐴)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ¬ 𝐷 ≼ 𝐴)
20		unxpwdom3.av	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
21	20	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐴 ∈ 𝑉)
22		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑏 → (𝑎 + 𝑑) = (𝑎 + 𝑏))
23	22	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → ((𝑎 + 𝑑) ∈ 𝐵 ↔ (𝑎 + 𝑏) ∈ 𝐵))
24	23	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (¬ (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ (𝑎 + 𝑏) ∈ 𝐵))
25	24	rspcv 3278	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐷 → (∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
26	25	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → (∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → ¬ (𝑎 + 𝑏) ∈ 𝐵))
27		unxpwdom3.ov	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐷) → (𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵))
28	27	3expa 1257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → (𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵))
29		elun 3715	. . . . . . . . . . . . 13 ⊢ ((𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵) ↔ ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
30	28, 29	sylib 207	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → ((𝑎 + 𝑏) ∈ 𝐴 ∨ (𝑎 + 𝑏) ∈ 𝐵))
31	30	orcomd 402	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → ((𝑎 + 𝑏) ∈ 𝐵 ∨ (𝑎 + 𝑏) ∈ 𝐴))
32	31	ord 391	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → (¬ (𝑎 + 𝑏) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
33	26, 32	syld 46	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ 𝑏 ∈ 𝐷) → (∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵 → (𝑎 + 𝑏) ∈ 𝐴))
34	33	impancom 455	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝑏 ∈ 𝐷 → (𝑎 + 𝑏) ∈ 𝐴))
35		unxpwdom3.lc	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))
36	35	ex 449	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ((𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
37	36	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → ((𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐)))
38	34, 37	dom2d 7882	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → (𝐴 ∈ 𝑉 → 𝐷 ≼ 𝐴))
39	21, 38	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵) → 𝐷 ≼ 𝐴)
40	19, 39	mtand 689	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ¬ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
41		dfrex2 2979	. . . . 5 ⊢ (∃𝑑 ∈ 𝐷 (𝑎 + 𝑑) ∈ 𝐵 ↔ ¬ ∀𝑑 ∈ 𝐷 ¬ (𝑎 + 𝑑) ∈ 𝐵)
42	40, 41	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ∃𝑑 ∈ 𝐷 (𝑎 + 𝑑) ∈ 𝐵)
43	17, 42	reximddv 3001	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ∃𝑑 ∈ 𝐷 ∃𝑦 ∈ 𝐵 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦))
44		vex 3176	. . . . . . . . 9 ⊢ 𝑑 ∈ V
45		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
46	44, 45	op1std 7069	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → (1st ‘𝑥) = 𝑑)
47	46	oveq2d 6565	. . . . . . 7 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑐 + (1st ‘𝑥)) = (𝑐 + 𝑑))
48	44, 45	op2ndd 7070	. . . . . . 7 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → (2nd ‘𝑥) = 𝑦)
49	47, 48	eqeq12d 2625	. . . . . 6 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → ((𝑐 + (1st ‘𝑥)) = (2nd ‘𝑥) ↔ (𝑐 + 𝑑) = 𝑦))
50	49	riotabidv 6513	. . . . 5 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → (℩𝑐 ∈ 𝐶 (𝑐 + (1st ‘𝑥)) = (2nd ‘𝑥)) = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦))
51	50	eqeq2d 2620	. . . 4 ⊢ (𝑥 = ⟨𝑑, 𝑦⟩ → (𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + (1st ‘𝑥)) = (2nd ‘𝑥)) ↔ 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦)))
52	51	rexxp 5186	. . 3 ⊢ (∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + (1st ‘𝑥)) = (2nd ‘𝑥)) ↔ ∃𝑑 ∈ 𝐷 ∃𝑦 ∈ 𝐵 𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + 𝑑) = 𝑦))
53	43, 52	sylibr 223	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → ∃𝑥 ∈ (𝐷 × 𝐵)𝑎 = (℩𝑐 ∈ 𝐶 (𝑐 + (1st ‘𝑥)) = (2nd ‘𝑥)))
54	4, 53	wdomd 8369	1 ⊢ (𝜑 → 𝐶 ≼* (𝐷 × 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ≼ cdom 7839   ≼^* cwdom 8345

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-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-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-wdom 8347

This theorem is referenced by:  isnumbasgrplem2  36693