Theorem infxp 10283

Description: Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
infxp	⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))

Proof of Theorem infxp

Step	Hyp	Ref	Expression
1		sdomdom 9040	. . 3 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴)
2		infxpabs 10280	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3		infunabs 10275	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
4	3	3expa 1118	. . . . . . . 8 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
5	4	adantrl 715	. . . . . . 7 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 ∪ 𝐵) ≈ 𝐴)
6	5	ensymd 9065	. . . . . 6 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ (𝐴 ∪ 𝐵))
7		entr 9066	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≈ 𝐴 ∧ 𝐴 ≈ (𝐴 ∪ 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))
8	2, 6, 7	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))
9	8	expr 456	. . . 4 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵 ≼ 𝐴 → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)))
10	9	adantrl 715	. . 3 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵 ≼ 𝐴 → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)))
11	1, 10	syl5 34	. 2 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵 ≺ 𝐴 → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)))
12		domtri2 10058	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
13	12	ad2ant2r 746	. . 3 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
14		xpcomeng 9130	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
15	14	ad2ant2r 746	. . . . 5 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
16		simplrl 776	. . . . . . 7 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → 𝐵 ∈ dom card)
17		domtr 9067	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → ω ≼ 𝐵)
18	17	ad4ant24 753	. . . . . . 7 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → ω ≼ 𝐵)
19		infn0 9368	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
20	19	ad3antlr 730	. . . . . . 7 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → 𝐴 ≠ ∅)
21		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐵)
22		infxpabs 10280	. . . . . . 7 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (𝐴 ≠ ∅ ∧ 𝐴 ≼ 𝐵)) → (𝐵 × 𝐴) ≈ 𝐵)
23	16, 18, 20, 21, 22	syl22anc 838	. . . . . 6 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → (𝐵 × 𝐴) ≈ 𝐵)
24		uncom 4181	. . . . . . . 8 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
25		infunabs 10275	. . . . . . . . 9 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵 ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
26	16, 18, 21, 25	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → (𝐵 ∪ 𝐴) ≈ 𝐵)
27	24, 26	eqbrtrid 5201	. . . . . . 7 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
28	27	ensymd 9065	. . . . . 6 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → 𝐵 ≈ (𝐴 ∪ 𝐵))
29		entr 9066	. . . . . 6 ⊢ (((𝐵 × 𝐴) ≈ 𝐵 ∧ 𝐵 ≈ (𝐴 ∪ 𝐵)) → (𝐵 × 𝐴) ≈ (𝐴 ∪ 𝐵))
30	23, 28, 29	syl2anc 583	. . . . 5 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → (𝐵 × 𝐴) ≈ (𝐴 ∪ 𝐵))
31		entr 9066	. . . . 5 ⊢ (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴 ∪ 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))
32	15, 30, 31	syl2an2r 684	. . . 4 ⊢ ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))
33	32	ex 412	. . 3 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 ≼ 𝐵 → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)))
34	13, 33	sylbird 260	. 2 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (¬ 𝐵 ≺ 𝐴 → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)))
35	11, 34	pm2.61d 179	1 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2946   ∪ cun 3974  ∅c0 4352   class class class wbr 5166   × cxp 5698  dom cdm 5700  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-dju 9970  df-card 10008

This theorem is referenced by:  alephmul  10647