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Theorem infxp 10210
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 8976 . . 3 (𝐵𝐴𝐵𝐴)
2 infxpabs 10207 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3 infunabs 10202 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
433expa 1119 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
54adantrl 715 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴𝐵) ≈ 𝐴)
65ensymd 9001 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → 𝐴 ≈ (𝐴𝐵))
7 entr 9002 . . . . . 6 (((𝐴 × 𝐵) ≈ 𝐴𝐴 ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
82, 6, 7syl2anc 585 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
98expr 458 . . . 4 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
109adantrl 715 . . 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 9984 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1312ad2ant2r 746 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 xpcomeng 9064 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1514ad2ant2r 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 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
17 domtr 9003 . . . . . . . 8 ((ω ≼ 𝐴𝐴𝐵) → ω ≼ 𝐵)
1817ad4ant24 753 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → ω ≼ 𝐵)
19 infn0 9307 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
2019ad3antlr 730 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴 ≠ ∅)
21 simpr 486 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴𝐵)
22 infxpabs 10207 . . . . . . 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 4154 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
25 infunabs 10202 . . . . . . . . 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 5184 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
2827ensymd 9001 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ≈ (𝐴𝐵))
29 entr 9002 . . . . . 6 (((𝐵 × 𝐴) ≈ 𝐵𝐵 ≈ (𝐴𝐵)) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
3023, 28, 29syl2anc 585 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
31 entr 9002 . . . . 5 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3215, 30, 31syl2an2r 684 . . . 4 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3332ex 414 . . 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 205  wa 397  wcel 2107  wne 2941  cun 3947  c0 4323   class class class wbr 5149   × cxp 5675  dom cdm 5677  ωcom 7855  cen 8936  cdom 8937  csdm 8938  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-dju 9896  df-card 9934
This theorem is referenced by:  alephmul  10573
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