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Theorem infxp 9022
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 7968 . . 3 (𝐵𝐴𝐵𝐴)
2 infxpabs 9019 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3 infunabs 9014 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
433expa 1263 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
54adantrl 751 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴𝐵) ≈ 𝐴)
65ensymd 7992 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → 𝐴 ≈ (𝐴𝐵))
7 entr 7993 . . . . . 6 (((𝐴 × 𝐵) ≈ 𝐴𝐴 ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
82, 6, 7syl2anc 692 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
98expr 642 . . . 4 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
109adantrl 751 . . 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 8800 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1312ad2ant2r 782 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 xpcomeng 8037 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1514ad2ant2r 782 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1615adantr 481 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
17 simplrl 799 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
18 simplr 791 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → ω ≼ 𝐴)
19 domtr 7994 . . . . . . . 8 ((ω ≼ 𝐴𝐴𝐵) → ω ≼ 𝐵)
2018, 19sylan 488 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → ω ≼ 𝐵)
21 infn0 8207 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
2221ad3antlr 766 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴 ≠ ∅)
23 simpr 477 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴𝐵)
24 infxpabs 9019 . . . . . . 7 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (𝐴 ≠ ∅ ∧ 𝐴𝐵)) → (𝐵 × 𝐴) ≈ 𝐵)
2517, 20, 22, 23, 24syl22anc 1325 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ 𝐵)
26 uncom 3749 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
27 infunabs 9014 . . . . . . . . 9 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2817, 20, 23, 27syl3anc 1324 . . . . . . . 8 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2926, 28syl5eqbr 4679 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
3029ensymd 7992 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ≈ (𝐴𝐵))
31 entr 7993 . . . . . 6 (((𝐵 × 𝐴) ≈ 𝐵𝐵 ≈ (𝐴𝐵)) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
3225, 30, 31syl2anc 692 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
33 entr 7993 . . . . 5 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3416, 32, 33syl2anc 692 . . . 4 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3534ex 450 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3613, 35sylbird 250 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (¬ 𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3711, 36pm2.61d 170 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 196  wa 384  wcel 1988  wne 2791  cun 3565  c0 3907   class class class wbr 4644   × cxp 5102  dom cdm 5104  ωcom 7050  cen 7937  cdom 7938  csdm 7939  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-card 8750  df-cda 8975
This theorem is referenced by:  alephmul  9385
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