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Theorem ismgmOLD 32622
Description: Obsolete version of ismgm 17012 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ismgmOLD.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
ismgmOLD (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Proof of Theorem ismgmOLD
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5925 . . . . 5 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
21exbidv 1836 . . . 4 (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3 df-mgmOLD 32621 . . . 4 Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
42, 3elab2g 3321 . . 3 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5 f00 5985 . . . . . . . 8 (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6 dmeq 5233 . . . . . . . . . 10 (𝐺 = ∅ → dom 𝐺 = dom ∅)
7 dmeq 5233 . . . . . . . . . . 11 (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8 dm0 5247 . . . . . . . . . . . . 13 dom ∅ = ∅
98dmeqi 5234 . . . . . . . . . . . 12 dom dom ∅ = dom ∅
109, 8eqtri 2631 . . . . . . . . . . 11 dom dom ∅ = ∅
117, 10syl6req 2660 . . . . . . . . . 10 (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
126, 11syl 17 . . . . . . . . 9 (𝐺 = ∅ → ∅ = dom dom 𝐺)
1312adantr 479 . . . . . . . 8 ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
145, 13sylbi 205 . . . . . . 7 (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15 xpeq12 5048 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
1615anidms 674 . . . . . . . . 9 (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17 feq23 5928 . . . . . . . . 9 (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
1816, 17mpancom 699 . . . . . . . 8 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
19 eqeq1 2613 . . . . . . . 8 (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
2018, 19imbi12d 332 . . . . . . 7 (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
2114, 20mpbiri 246 . . . . . 6 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
22 fdm 5950 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23 dmeq 5233 . . . . . . . 8 (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24 df-ne 2781 . . . . . . . . . . . 12 (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25 dmxp 5252 . . . . . . . . . . . 12 (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
2624, 25sylbir 223 . . . . . . . . . . 11 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
2726eqeq1d 2611 . . . . . . . . . 10 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺𝑡 = dom dom 𝐺))
2827biimpcd 237 . . . . . . . . 9 (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
2928eqcoms 2617 . . . . . . . 8 (dom dom 𝐺 = dom (𝑡 × 𝑡) → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3022, 23, 293syl 18 . . . . . . 7 (𝐺:(𝑡 × 𝑡)⟶𝑡 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3130com12 32 . . . . . 6 𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
3221, 31pm2.61i 174 . . . . 5 (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺)
3332pm4.71ri 662 . . . 4 (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
3433exbii 1763 . . 3 (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
354, 34syl6bb 274 . 2 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡)))
36 dmexg 6966 . . 3 (𝐺𝐴 → dom 𝐺 ∈ V)
37 dmexg 6966 . . 3 (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38 xpeq12 5048 . . . . . . 7 ((𝑡 = dom dom 𝐺𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
3938anidms 674 . . . . . 6 (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40 feq23 5928 . . . . . 6 (((𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑡 = dom dom 𝐺) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
4139, 40mpancom 699 . . . . 5 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
42 ismgmOLD.1 . . . . . . . 8 𝑋 = dom dom 𝐺
4342eqcomi 2618 . . . . . . 7 dom dom 𝐺 = 𝑋
4443, 43xpeq12i 5051 . . . . . 6 (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
4544, 43feq23i 5938 . . . . 5 (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋)
4641, 45syl6bb 274 . . . 4 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
4746ceqsexgv 3304 . . 3 (dom dom 𝐺 ∈ V → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4836, 37, 473syl 18 . 2 (𝐺𝐴 → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4935, 48bitrd 266 1 (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wne 2779  Vcvv 3172  c0 3873   × cxp 5026  dom cdm 5028  wf 5786  Magmacmagm 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794  df-mgmOLD 32621
This theorem is referenced by:  clmgmOLD  32623  opidonOLD  32624  issmgrpOLD  32635
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