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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1t1e1ALT | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
1t1e1ALT | ⊢ (1 · 1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11211 | . 2 ⊢ 1 ∈ ℝ | |
2 | ax-1rid 11177 | . 2 ⊢ (1 ∈ ℝ → (1 · 1) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7406 ℝcr 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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