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Theorem infxp 8892
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 7841 . . 3 (𝐵𝐴𝐵𝐴)
2 infxpabs 8889 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
3 infunabs 8884 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
433expa 1256 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
54adantrl 747 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴𝐵) ≈ 𝐴)
65ensymd 7865 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → 𝐴 ≈ (𝐴𝐵))
7 entr 7866 . . . . . 6 (((𝐴 × 𝐵) ≈ 𝐴𝐴 ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
82, 6, 7syl2anc 690 . . . . 5 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
98expr 640 . . . 4 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝐵 ≠ ∅) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
109adantrl 747 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
111, 10syl5 33 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
12 domtri2 8670 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1312ad2ant2r 778 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 xpcomeng 7909 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1514ad2ant2r 778 . . . . . 6 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
1615adantr 479 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
17 simplrl 795 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
18 simplr 787 . . . . . . . 8 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → ω ≼ 𝐴)
19 domtr 7867 . . . . . . . 8 ((ω ≼ 𝐴𝐴𝐵) → ω ≼ 𝐵)
2018, 19sylan 486 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → ω ≼ 𝐵)
21 infn0 8079 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
2221ad3antlr 762 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴 ≠ ∅)
23 simpr 475 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐴𝐵)
24 infxpabs 8889 . . . . . . 7 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (𝐴 ≠ ∅ ∧ 𝐴𝐵)) → (𝐵 × 𝐴) ≈ 𝐵)
2517, 20, 22, 23, 24syl22anc 1318 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ 𝐵)
26 uncom 3713 . . . . . . . 8 (𝐴𝐵) = (𝐵𝐴)
27 infunabs 8884 . . . . . . . . 9 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2817, 20, 23, 27syl3anc 1317 . . . . . . . 8 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵𝐴) ≈ 𝐵)
2926, 28syl5eqbr 4607 . . . . . . 7 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
3029ensymd 7865 . . . . . 6 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → 𝐵 ≈ (𝐴𝐵))
31 entr 7866 . . . . . 6 (((𝐵 × 𝐴) ≈ 𝐵𝐵 ≈ (𝐴𝐵)) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
3225, 30, 31syl2anc 690 . . . . 5 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐵 × 𝐴) ≈ (𝐴𝐵))
33 entr 7866 . . . . 5 (((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ≈ (𝐴𝐵)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3416, 32, 33syl2anc 690 . . . 4 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) ∧ 𝐴𝐵) → (𝐴 × 𝐵) ≈ (𝐴𝐵))
3534ex 448 . . 3 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴𝐵 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3613, 35sylbird 248 . 2 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (¬ 𝐵𝐴 → (𝐴 × 𝐵) ≈ (𝐴𝐵)))
3711, 36pm2.61d 168 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 194  wa 382  wcel 1975  wne 2774  cun 3532  c0 3868   class class class wbr 4572   × cxp 5021  dom cdm 5023  ωcom 6929  cen 7810  cdom 7811  csdm 7812  cardccrd 8616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-oi 8270  df-card 8620  df-cda 8845
This theorem is referenced by:  alephmul  9251
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