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Theorem infxp 10255
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) ≈ (𝐴𝐵))

Proof of Theorem infxp
StepHypRef Expression
1 sdomdom 9021 . . 3 (𝐵𝐴𝐵𝐴)
2 infxpabs 10252 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3 infunabs 10247 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
433expa 1118 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
54adantrl 716 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴𝐵) ≈ 𝐴)
65ensymd 9046 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → 𝐴 ≈ (𝐴𝐵))
7 entr 9047 . . . . . 6 (((𝐴 × 𝐵) ≈ 𝐴𝐴 ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
82, 6, 7syl2anc 584 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
98expr 456 . . . 4 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
109adantrl 716 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
111, 10syl5 34 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
12 domtri2 10030 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1312ad2ant2r 747 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 xpcomeng 9105 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1514ad2ant2r 747 . . . . 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 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
17 domtr 9048 . . . . . . . 8 ((ω ≼ 𝐴𝐴𝐵) → ω ≼ 𝐵)
1817ad4ant24 754 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → ω ≼ 𝐵)
19 infn0 9341 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
2019ad3antlr 731 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴 ≠ ∅)
21 simpr 484 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴𝐵)
22 infxpabs 10252 . . . . . . 7 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (𝐴 ≠ ∅ ∧ 𝐴𝐵)) → (𝐵 × 𝐴) ≈ 𝐵)
2316, 18, 20, 21, 22syl22anc 838 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ 𝐵)
24 uncom 4157 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
25 infunabs 10247 . . . . . . . . 9 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2616, 18, 21, 25syl3anc 1372 . . . . . . . 8 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2724, 26eqbrtrid 5177 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
2827ensymd 9046 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ≈ (𝐴𝐵))
29 entr 9047 . . . . . 6 (((𝐵 × 𝐴) ≈ 𝐵𝐵 ≈ (𝐴𝐵)) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
3023, 28, 29syl2anc 584 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
31 entr 9047 . . . . 5 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3215, 30, 31syl2an2r 685 . . . 4 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3332ex 412 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3413, 33sylbird 260 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (¬ 𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3511, 34pm2.61d 179 1 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2107  wne 2939  cun 3948  c0 4332   class class class wbr 5142   × cxp 5682  dom cdm 5684  ωcom 7888  cen 8983  cdom 8984  csdm 8985  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-dju 9942  df-card 9980
This theorem is referenced by:  alephmul  10619
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