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Theorem 1t1e1ALT 41174
Description: Alternate proof of 1t1e1 12371 using a different set of axioms (add ax-mulrcl 11170, ax-i2m1 11175, ax-1ne0 11176, ax-rrecex 11179 and remove ax-resscn 11164, ax-mulcom 11171, ax-mulass 11173, ax-distr 11174). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
1t1e1ALT (1 · 1) = 1

Proof of Theorem 1t1e1ALT
StepHypRef Expression
1 1re 11211 . 2 1 ∈ ℝ
2 ax-1rid 11177 . 2 (1 ∈ ℝ → (1 · 1) = 1)
31, 2ax-mp 5 1 (1 · 1) = 1
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  (class class class)co 7406  cr 11106  1c1 11108   · cmul 11112
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-ext 2704  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-mulcl 11169  ax-mulrcl 11170  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rrecex 11179  ax-cnre 11180
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-ov 7409
This theorem is referenced by:  nnmul1com  41183  remulinvcom  41302  sn-0tie0  41309
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